imme van den berg vitor neves (eds.)
.fl SpringerWienNewYork
fl
SpringerWienNewYork
Imme van den Berg Vitor Neves (eds.) The Strength of Nonstandard Analysis
SpringerWienNewYork
Imme van den Berg Departamento de Matematica Universidade de Evora, Evora, Portugal
Vitor Neves Departamento de Matematica Universidade de Aveiro, Aveiro, Portugal
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Foreword Willst du ins Unendliche schreiten? Gch nur im Endlichcn nach allc Scitcn! Willst du dich am Ganzen erquicken, So must du das Ganze im Kleinsten erblicken.
J. W. Goethe
(Gott, Gemut
und
Welt, 1815)
Fortyfive years ago, an article appeared in the Proceedings of the Royal Academy of Sciences of the Netherlands Series A, 64, 432440 and lndagationes Math. 23 ( 4) , 196 1 , with the mysterious title "Nonstandard Analysis" authored by the eminent mathematician and logician Abraham Robinson ( 1 908 1974) . The title of the paper turned out to be a contraction of the two terms "Non standard Model" used in model theory and "Analysis". It presents a treatment of classical analysis based on a theory of infinitesimals in the context of a nonstandard model of the real number system R In the Introduction of the article, Robinson states: "It is our main purpose to show that the models provide a natural approach to the age old problem of producing a calculus involving infinitesimal (infinitely small) and infinitely large quantities. As is wellknown the use of infinitesimals strongly advocated by Leibniz and unhesitatingly accepted by Euler fell into disrepute after the advent of Cauchy ' s methods which put Mathematical Analysis on a firm foundation". To bring out more clearly the importance of Robinson ' s creation of a rigor ous theory of infinitesimals and their reciprocals, the infinitely large quantities, that has changed the landscape of analysis, I will briefly share with the reader a few highlights of the historical facts that arc involved. The invention of the "Infinitesimal Calculus" in the second half of the sev enteenth century by Newton and Lcibniz can be looked upon as the first funda
vi
Foreword
mental new discovery in mathematics of revolutionary nature since the death of Archimedes in 2 1 2 BC. The fundamental discovery that the operations of differentiation (flux) and integration (sums of infinitesimal increments) are inverse operations using the intuitive idea that infinitesimals of higher order compared to those of lower order may be neglected became an object of severe criticism. In the "Analyst", section 35, Bishop G. Berkeley states: "And what are these fluxions? The velocities of evanescent incre ments. And what are these same evanescent increments? They are neither finite quantities, nor quantities infinitesimally small, nor yet nothing. May we call them ghosts of departed quantities?" The unrest and criticism concerning the lack of a rigorous foundation of the in finitesimal calculus led the Academy of Sciences of Berlin, at its public meeting on June 3, 1774, and well on the insistence of the Head of the rviathematics Sec tion, .J. L. Lagrange, to call upon the mathematical community to solve this important problem. To this end, it announced a prize contest dealing with the problem of "Infinity" in the broadest sense possible in mathematics. The announcement read: "The utility derived from Mathematics, the esteem it is held in and the honorable name of 'exact science' par excellence, that it justly deserves, are all due to the clarity of its principles, the rigor of its proofs and the precision of its theorems. In order to ensure the continuation of these valuable attributes in this important part of our knowledge the prize of a 50 ducat gold medal is for: A clear and precise theory of what is known as 'Infinity' in Mathe matics. It is wellknown that higher mathematics regularly makes use of the infinitely large and infinitely small. The geometers of antiquity and even the ancient analysts, however, took great pains to avoid anything approaching the infinity, whereas today's emi nent modern analysts admit to the statement 'infinite magnitude' is a contradiction in terms. For this reason the Academy desires an explanation why it is that so many correct theorems have been deduced from a contradictory assumption, together with a formula tion of a truly clear mathematical principle that may replace that of infinity without , however, rendering investigations by its usc overly difficult and overly lengthy. It is requested that the sub ject be treated in all possible generality and with all possible rigor, clarity and simplicity."
Foreword
vii
Twentythree answers were received before the deadline of January 1 , 1 786. The prize was awarded to the Swiss mathematician Simon L'Huilier for his essay with motto: "The infinite is the abyss in which our thoughts are engulfed." The members of the "Prize Committee" made the following noteworthy points: None of the submitted essays dealt with the question raised "why so many correct theorems have been derived from a contradictory assumption?" Furthermore , the request for clarity, simplicity and, above all , rigor was not met by the contenders, and almost all of them did not address the request for a newly formulated principle of infinity that would reach beyond the infinitesimal calculus to be meaningful also for algebra and geometry. For a detailed account of the prize contest we refer the reader to the interest ing biography of Lazare Nicolas M. Carnot ( 1 7531823) , the father of the ther modynamicist Sadi Carnot , entitled "Lazare Carnot Savant" by Ch. C . Gille spie ( Princeton Univ. Press, 1971 ) , which contains a thorough discussion of Carnot's entry "Dissertation sur la theorie de l'infini mathematique", received by the Academy after the deadline. The above text of the query was adapted from the biography. In retrospect , the outcome of the contest is not surprising. Nevertheless around that time the understanding of infinitesimals had reached a more so phisticated level as the books of J. L. Lagrange and L. N. Carnot published in Paris in 1797 show. From our present state of the art, it seems that the natural place to look for a "general principle of infinity" is set theory. Not however for an "intrinsic" definition of infinity. Indeed, as Gian Carlo Rota expressed not too long ago: "God created infinity and man, unable to understand it , had to invent finite sets." At this point let me digress a little for further clarification about the infinity we are dealing with. During the early development of Cantor's creation of set theory, it was E. Zermelo who realized that the attempts to prove the existence of "infinite" sets, short of assuming there is an "infinite" set or a nonfinite set as in Proposition 66 of Dedekind's famous "Was sind und was sollen die Zahlen?" were fallacious. For this reason, Zermelo in his important paper "Sur les ensembles finis et le principe de !'induction complete", Acta Math. 32 ( 1909) , 185193 ( submitted in 1907) , introduced an axiom of"infinity" by postulating the existence of a set , say A , nonempty, and that for each of its elements x, the singleton { x} is an element of it. Returning to the request of the Academy: To discover a property that all infinite sets would have in common with the finite sets that would facilitate
viii
Foreword
their use in all branches of mathematics. What comes to mind is Zermelo's well ordering principle. Needless to say that this principle and the manifold results and consequences in all branches of mathematics have had an enormous impact on the development of mathematics since its introduction. One may ask what has this to do with the topic at hand? It so happens that the existence of nonstandard models depends essentially on it as well and consequently non standard analysis too. The construction of the real number system (linear continuum) by Cantor and Dedekind in 1872 and the Weierstrass c:0 technique gradually replaced the use of infinitesimals. Hilbert's characterization in 1899 of the real num ber system as a (Dedekind) complete field led to the discovery, in 1907, by H. Hahn, of nonarchimedian totally ordered field extensions of the reals. This development brought about a renewed interest in the theory of infinitesimals. The resulting "calculus", certainly of interest by itself, lacked a process of defin ing extensions of the elementary and special functions, etc. , of the objects of classical analysis. It is interesting that Cantor strongly rejected the existence of nonarchimedian totally ordered fields. He expressed the view that no ac tual infinities could exist other than his transfinite cardinal numbers and that, other than 0 , infinitesimals did not exist . He also offered a "proof" in which he actually assumed order completeness. It took one hundred and seventyfive years from the time of the deadline of the Berlin Academy contest to the publication of Robinson's paper "Non standard Analysis". As Robinson told us, his discovery did not come about as a result of his efforts to solve Leibniz ' problem; far from it . Working on a paper on formal languages where the length of the sentences could be countable, it occurred to him to look up again the important paper by T. Skolem ''tJber die Nichtcharakterisierbarkeit der Zahlenreihe mittcls endlich oder abzahlbar unendlich vieler Aussagen mit ausschliesslich Zahlenvariablen, Fund. Math. 23 ( 1 934) , 1 50161 * . Briefly, Skolem showed in his paper the existence of models of Peano arith metic having "infinitely large numbers". Nevertheless in his models the prin ciple of induction holds only for subsets determined by admissible formulas from the chosen formal language used to describe Peano's axiom system. The nonempty set of the infinitely large numbers has no smallest element and so cannot be determined by a formula of the formal language and is called an external set; those that can were baptized as internal sets of the model. Robinson, rereading Skolem's paper, wondered what systems of numbers would emerge if he would apply Skolem 's method to the axiom system of the real numbers . In doing so, Robinson immediately realized that the real number *
sec abo: T. Skolcm "Pcano'� Axiom� and Modcb of Arithmetic". in Sympo�ium on the
Mathematical Interpretation of Formal Systems, NorthHolland, Amsterdam
1955, 114.
Foreword
ix
system was a nonarchimedian totally ordered field extension of the reals whose structure satisfies all the properties of the reals, and that , in particular, the set of infinitesimals lacking a least upper bound was an external set. This is how it all started and the Academy would certainly award Robinson the gold medal. At the end of the fifties at Caltech (California Institute of Technology) Arthur Erdelyi FRS ( 1908 1977) conducted a lively seminar entitled "Gener alized Functions". It dealt with various areas of current research at that time in such fields as J. Mikusinski's rigorous foundation of the socalled Heaviside operational calculus and L. Schwartz' t heory of distributions. In connection with Schwartz' distribution theory, Erdelyi urged us to read the just appeared papers by Laugwitz and Schmieden dealing with the representations of the Diracdelta functions by sequences of pointfunctions converging to 0 point wise except at 0 where they run to infinity. Robinson's paper fully clarified this phenomenon. Reduced powers of � instead of ultrapowers, as in Robin son's paper, were at play here. In my 1962 Notes on Nonstandard Analysis the ultrapower construction was used, but at that time without using explicitly the Transfer Principle. In 1967 the first International Symposium on Nonstandard Analysis took place at Caltech with the support of the U.S. Office of Naval Research. At the time the use of nonstandard models in other branches of mathematics started to blossom. This is the reason that the Proceedings of the Symposium carries the title: Applications of Model Theory to Algebra. Analysis and Probability. A little anecdote about the meeting. When I opened the newspaper one morning during the week of the meeting, I discovered to my surprise that it had attracted the attention of the Managing Editor of the Pasadena Star News; his daily "Conversation Piece" read: "A Stanford Professor spoke in Pasadena this week on the subject 'Axiomatizations of Nonstandard Analysis which are Conservative Extensions of Formal Systems for Standard Classical Analysis', a fact which I shall tuck away for reassurance on those days when I despair of communicating clearly." may add here that from the beginning Robinson was very interested in the formulation of an axiom system catching his nonstandard methodology. Unfortunately he did not live to see the solution of his problem by E. Nelson presented in the 1977 paper entitled "Internal Set Theory". A presentation by Nelson, "The virtue of Simplicity", can be found in this book. A final observation. During the last sixty years we have all seen come about the solutions of a number of outstanding problems and conjectures,
X
Foreword
some centuries old, that have enriched mathematics. The centuryold problem to create a rigorous theory of infinitesimals no doubt belongs in this category. It is somewhat surprising that the appreciation of Robinson's creation was slow in coming. Is it possible that the finding of the solution in model theory, a branch of mathematical logic, had something to do with that? The answer may perhaps have been given by Augustus de Morgan ( 1 8061871 ) , who is wellknown from De 1Iorgan's Law , and who in collabo ration with George Boolc ( 1 805 1864) reestablished formal logic as a branch of exact science in the nineteenth century, when he wrote: "We know that mathematicians care no more for logic than logicians for mathematics. The two eyes of exact sciences are mathematics and logic: the mathematical sect puts out the logical eye, the logical sect puts out the mathematical eye; each believing that it can sec better with one eye than with two." We owe Abraham Robinson a great deal for having taught us the use of both eyes. This book shows clearly that we have learned our lesson well. All the contributors are to be commended for the way they have made an effort to make their contributions that are based on the talks at the meeting "Nonstandard Mathematics 2004" as selfcontained as can be expected. For further facilitating the readers, the editors have divided the papers in categories according to the subject. The whole presents a very rich assortment of the non standard approach to diverse areas of mathematical analysis. I wish it many readers. Wilhelmus A. .J. Luxemburg Pasadena, California September 2006
Acknowledgements
The wide range of applicability of Mathematical Logic to classical Mathe matics, beyond Analysis  as Robinson's terminology Nonstandard Analysis might imply  is already apparent in the very first symposium on the area, held in Pasadena in 1967, as mentioned in the foreword. Important indicators of the maturity of this field are the high level of foun dational and pure or applied mathematics presented in this book, congresses which take place approximately every two years , as well as the experiences of teaching, which proliferated at graduate , undergraduate and secondary school level all over the world. This book, made of peer reviewed contributions, grew out of the meet ing Non Standard Mathematics 2004, which took place in .July 2004 at the Department of Mathematics of the University of Aveiro (Portugal) . The ar ticles are organized into five groups, ( 1 ) Foundations, (2) Number theory, (3) Statistics, probability and measures, ( 4) Dynamical systems and equations and (5) lnfinitcsimals and education. Its cohesion is enhanced by many cross overs. We thank the authors for the care they took with their contributions as well as the three Portuguese Research Institutes which funded the production cost: the Centro lnternacional de Matematica CIM at Coimbra, the Centro de Estudos em Optimiza.:;ao e Controlo CEO C of the University of Aveiro and the Centro de lnvestiga.:;ao em Matematica e Aplica.:;oes CIMAUE of the University of Evora. Moreover, NS1I2004 would not have taken place were it not for the support of the CIM, CEOC and CIMA themselves , Funda.:;ao LusoAmericana para o Desenvolvimento, the Centro de Analise Matematica, Geomctria e Sistemas Dinamicos of the lnstituto Superior Tccnic:o of Lisbon and the Mathematics Department at Aveiro. Our special thanks go to Wilhclmus Luxemburg of Caltech, USA, who, being one of the first to recognize the importance of Robinson's discovery of
xii
Acknowledgements
Nonstandard Analysis , honoured us by writing the foreword. Last but not least , Salvador SanchezPedreiio of the University of Murcia, Spain, joined his expert knowledge of Nonstandard Analysis to a high dominion of the 11\'JEX editing possibilities in order to improve the consistency of the text and to obtain an outstanding typographic result. Imme van den Berg Vftor Neves Editors
Contents I
Foundations
1 The strength of nonstandard analysis by H . .JEROME KEISLER 1.1 Introduction . . . . . 1.2 The theory PRAw . . 1.3 The theory NPRAw . 1.4 The theory WNA . . 1.5 Bounded minima and overs pill . 1.6 Standard parts . . . . . . . . 1.7 Liftings of formulas . . . . . . 1.8 Choice principles in L(PRAw) 1 .9 Saturation principles . . . . 1 . 1 0 Saturation and choice 1 . 1 1 Second order standard parts 1 . 1 2 Functional choice and (3 2 ) 1 . 13 Conclusion . . . . . . . 2 The virtue of simplicity by EDWARD NELSON Part I. Technical Part II. General .
1
3 3 4 6 7 9 11 14 16 17 20 21 23 25 27 27 30
3 Analysis of various practices of referring in classical or non standard mathematics by YVES PERAIRE 3. 1 Introduction . . . . . . . . . . . . . . . 3. 2 Generalites sur la referentiation . . . . 3.3 Le calcul de Dirac. L'egalite de Dirac . 3.4 Calcul de Heaviside sans transformee de Laplace. L'egalite de Laplace . 3 . 5 Exemples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33 33 34 37 41 44
xiv
Contents
4 Stratified analysis? by KAREL H RBACEK 4.1 The Robinsonian framework . . . . . . . . . . 4.2 Stratified analysis . . . . . . . . . . . . . . . . 4.3 An axiomatic system for stratified set theory
47
5 ERNA at work by C . IMPENS and S . SANDERS 5.1 Introduction . . . . . . 5.2 The system . . . . . . 5.2.1 The language 5.2.2 The axioms . .
64
6 The Sousa Pinto approach to nonstandard generalised functions by R. F. HOSKINS 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Generalised functions and N.S.A. . . . . . . 6.2 Distributions, ultradistributions and hyperfunctions . 6.2.1 Schwartz distributions . . . . . . . . . . . 6.2.2 The Silva axioms 6.2.3 Fourier transforms and ultradistributions 6.2.4 Sato hyperfunctions . . . . . . . . . . . . 6.2.5 Harmonic representation of hyperfunctions 6.3 Prehyperfunctions and predistributions . 6.4 The differential algebra A(D,J ....... . 6.4.1 Predistributions of finite order . . . 6.4.2 Predistributions of local finite order Predistributions of infinite order 6.4.3 6.5 Conclusion . . . . . . . . . . . 7 N eutrices in more dimensions by IMME VAN DEN BERG 7.1 Introduction . . . . . . . . . . 7.1.1 Motivation and objective 7.1.2 Setting . . . . . . . . . 7.1.3 Structure of this article . 7.2 The decomposition theorem . . . 7.3 Geometry of neutrices in � 2 and proof of the decomposition theorem . . . . . . . . . . . . . . . . . . . . . . . Thickness, width and length of neutrices 7.3.1 On the division of neutrices 7.3.2 Proof of the decomposition theorem . . . 7.3.3
48 53 58
64 65 65 67 76 76 77 77 77 78 80 81 83 84 86 86 88 88 90 92 92 92 94 95 95 96 97 103 111
Contents
II
Number theory
8 Nonstandard methods for additive and combinatorial number theory. A survey by RENLING .liN 8.1 The beginning . . . . . . 8.2 Duality between null ideal and meager ideal 8.3 Buyonegetonefree scheme . . . . . . . . . 8.4 From Kneser to Banach . . . . . . . . . . . 8.5 Inverse problem for upper asymptotic density 8.6 Freiman's 3k 3 + b conjecture . . . . . . . .
XV
117 119 119 120 121 124 125 129
9 Nonstandard methods and the ErdosTuran conjecture by STEVEN C . LETH 9.1 Introduction . . . . . . . . . . . 9.2 Near arithmetic progressions . . 9.3 The intervalmeasure property .
133
III
143
Statistics, probability and measures
133 134 140
10 Nonstandard likeliho od ratio test in exponential families 145 by JACQUES BosciRAUD 10.1 Introduction . . . . . . . . . . . . . . . . . 145 145 10 .1.1 A most powerful nonstandard test 10.2 Some basic concepts of statistics 146 10.2.1 Main definitions 146 146 10.2.2 Tests . . . . . 10.3 Exponential families . . 147 147 10.3.1 Basic concepts . 148 10.3.2 KullbackLeibler information number 10.3.3 The nonstandard test . 149 150 10.3.4 Large deviations for X . . . . . . . . 151 10.3.5 nregular sets . . . . . . . . . . . . . 10.3.6 nregular sets defined by KullbackLeibler information . 156 159 10.4 The nonstandard likelihood ratio test . a 162 10.4 .1 lnn infinitesimal 10.4.2 0 � � l ln ex I �co . . . . . . . . 163 10.4.3 � l ln ex I� co . . . . . . . . . . . 164 10.5 Comparison with nonstandard tests based on X . 165 165 10.5.1 Regular non ;:;tandard tests 10.5.2 Case when 8o is convex . . . . . . . . . . 167
xvi
Contents
1 1 A finitary approach for the representation of the infinitesimal generator of a markovian semigroup by S CHERAZADE BENHABIB 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 11.2 Construction of the least upper bound of sums in 1ST 11.3 The global part of the infinitesimal generator 11.4 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 12 On two recent applications of nonstandard analysis to the theory of financial markets by FREDERIK S . HERZBERG 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 A fair price for a multiply traded asset . . . . . . . . . . 12.3 Fairnessenhancing effects of a currency transaction tax 12.4 How to minimize "unfairness" . . . . . . . . . . 13 Quantum Bernoulli experiments and quantum stochastic pro cesses by MANFRED WOLFF 13.1 Introduction . . . . . . . . . . . . . . 13.2 Abstract quantum probability spaces 13. 3 Quantum Bernoulli experiments . . . 13.4 The internal quantum processes . . . 13.5 From the internal to the standard world 13.5.1 Brownian motion . . . . . . . . 13.5.2 The nonstandard hulls of the basic internal processes 13.6 The symmetric Fock space and its embedding into £ . . . . . 14 Applications of rich measure spaces formed from nonstandard models by PETER LOEB 14.1 Introduction . . . . . . . . . . . . . . 14.2 Recent work of Yeneng Sun . . . . . 14.3 Purification of measurevalued maps 15 More on Smeasures by DAVID A . Ross 15.1 Introduction . . . . 15.2 Loeb measures and Smeasures 15.3 Egoroff"s Theorem . . . . . . . 15.4 A Theorem of Riesz . . . . . . 15.4.1 Conditional expectation .
170 170 172 174 176
177 177 178 180 181
189 189 191 192 194 196 197 198 202
206 206 207 210
2 17 217 217 221 222 225
Contents 16 A RadonNikodym theorem for a vectorvalued reference measure by G . BEATE ZIMMER 16.1 Introduction and notation 16.2 The existing literature . . 16.3 The nonstandard approach . 16.4 A nonstandard vectorvector integral 16.5 Uniform convexity . . . . . . . . . . 16.6 Vectorvector derivatives without uniform convexity . 16.7 Remarks . . . . . . . . . . . . . . 17 Differentiability of Loeb measures by EVA AIGNER 17.1 Introduction . . . . . . . . . . . . 17.2 Sdifferentiability of internal measures 17.3 Differentiability of Loeb measures . . .
xvii
227 227 228 230 232 233 234 236
238 238 240 243
Differential systems and equations
2 51
18 The power of Gateaux differentiability by VfTOR NEVES 18.1 Preliminaries . . . . . . . . . 18.2 Smoothness . . . . . . . . . . 18.3 Smoothness and finite points 18.4 Smoothness and the nonstandard hull 18.4.1 Strong uniform differentiability . 18.4.2 The nonstandard hull
253
19 Nonstandard PalaisSmale conditions by NATALIA MARTINS and VfTOR NEVES 19.1 Preliminaries . . . . . . . . . . . . . 19.2 The PalaisSmale condition . . . . . 19.3 Nonstandard PalaisSmale conditions 19.4 PalaisSmale conditions per level . . 19.5 Nonstandard variants of PalaisSmale conditions per level 19.6 Mountain Pass Theorems . . . . . . . . . . . . . .
271
IV
20 Averaging for ordinary differential equations and functional differential equations by TEWFIK SARI 20.1 Introduction . . . . . . . . . . .
253 257 261 263 264 266
271 273 274 280 281 282
286 286
xviii
Contents
20. 2 Deformations and perturbations . 20.2.1 Deformations . . . . . . . 20.2.2 Perturbations . . . . . . 20.3 Averaging in ordinary differential equations 20.3.1 KBM vector fields . . . . . . . . . . 20.3.2 Almost solutions . . . . . . . . . . . 20.3.3 The stroboscopic method for ODEs 20.3.4 Proof of Theorem 2 for almost periodic vector fields 20.3.5 Proof of Theorem 2 for KBM vector fields . . . . 20.4 Functional differential equations . . . . . . . . . . . . . . . 20.4.1 Averaging for FDEs in the formz1(T) = cj (T,zT) 20.4.2 The stroboscopic method for ODEs revisited . . . 20.4.3 Averaging for FDEs in the form x(t) = f ( t/ c , Xt ) 20.4.4 The stroboscopic method for FDEs . . . . . . . .
21 Pathspace measure for stochastic differential equation with a co efficient of polynomial growth by T ORU NAKAMURA 21.1 Heuristic arguments and definitions . . . . . . . . . . 21. 2 Bounds for the *measure and the *Green function . 21.3 Solution to the FokkerPlanck equation . . . . 22 Optimal control for NavierStokes equations by NIGEL J . CuTLAND and K ATARZYNA G RZESIAK 22.1 Introduction . . . . . . . . . . 22.2 Preliminaries . . . . . . . . . . . . . . . . . . . 22.2.1 Nonstandard analysis . . . . . . . . . . 22.2. 2 The stochastic NavierStokes equations 22.2.3 Controls . . . . . . . . . . 22.3 Optimal control for d = 2 . . . . . 22.3.1 Controls with no feedback 22.3.2 Costs . . . . . . . . . . . . 22.3.3 Solutions for internal controls 22.3.4 Optimal controls . . . . . . . . 22.3.5 Holder continuous feedback controls (d = 2) 22.3.6 Controls based on digital observations (d = 2) 22.3.7 The space 1i . . . . . . . . . . . . . . . . . . . 22.3.8 The observations . . . . . . . . . . . . . . . . . 22.3.9 Ordinary and relaxed feedback controls for digital observations . . . . . . . . . . . . . 22.3.10 Costs for digitally observed controls . . . . . . . .
287 287 288 290 291 292 294 294 295 297 298 299 300 301
306 306 309 310
3 17 317 319 319 319 323 325 325 326 327 328 329 331 331 332 332 334
Contents 22.3. 1 1 Solution of the equations 22.3. 12 Optimal control . . . . . 22.4 Optimal control for d = 3 . . . . 22.4. 1 Existence of solutions for any control 22.4.2 The control problem for 3D stochastic NavierStokes equations . . . . . . . . 22.4.3 The space 0 . . . . . . 22.4.4 Approximate solutions 22.4.5 Optimal control . . . . 22.4.6 Holder continuous feedback controls (d = 3) 22.4.7 Approximate solutions for Holder continuous controls Appendix: Nonstandard representations of the spaces Hr 23 Lo calintime existence of strong solutions of the ndimensional Burgers equation via discretizations by .JOAO PAULO TEIXEIRA 23. 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 23.2 A discretization for the diffusionadvection equations in the torus . . . . . . . . . . . . . . . . . . . . 23.3 Some standard estimates for the solution of the discrete problem . . . . . . . . . . . . . . . . . 23.4 Main estimates on the hyperfinite discrete problem 23.5 Existence and uniqueness of solution . . . . . . . .
V
Infinitesimals and education
24 Calculus with infinitesimals by KEITH D. STROYAN 24. 1 Intuitive proofs with "small" quantities 24. 1 . 1 Continuity and extreme values 24. 1 . 2 Microscopic tangency in one variable 24. 1 .3 The Fundamental Theorem of Integral Calculus 24. 1 . 4 Telescoping sums and derivatives . . . . . . . . 24. 1 . 5 Continuity of the derivative . . . . . . . . . . . . 24. 1 . 6 Trig, polar coordinates, and Holditch's formula . 24. 1 .7 The polar area differential . . . . . . . . 24. 1.8 Leibniz's formula for radius of curvature 24. 1 . 9 Changes . . . . . . . . . 24. 1 . 10 Small changes . . . . . . 24. 1 . 1 1 The natural exponential
xix 334 336 337 338 338 339 340 342 343 344 345 349 349 351 353 356 361 367 369 369 369 370 372 372 373 375 376 379 379 380 381
XX
Contents 24. 1 . 1 2 Concerning the history of the calculus . . . . 24.2 Keisler's axioms . . . . . . . . . . . . . . . . . . . . . 24.2. 1 Small, medium, and large hyperreal numbers 24.2.2 Keisler's algebra axiom . . . . . . 24.2.3 The uniform derivative of x 3 . . . 24.2.4 Keisler's function extension axiom 24.2.5 Logical real expressions 24.2.6 Logical real formulas . . . . . 24.2. 7 Logical real statements . . . . 24.2.8 Continuity and extreme values 24.2.9 Microscopic tangency in one variable 24. 2 . 1 0 The Fundamental Theorem of Integral Calculus 24.2 . 1 1 The Local Inverse Function Theorem . . . . . . 24. 2 . 1 2 Second differences and higher order smoothness
25 PreUniversity Analysis by RICHARD O' DONOVAN 25. 1 Introduction . . . . 25.2 Standard part . . . 25.3 Stratified analysis 25.4 Derivative . . . . . 25.5 Transfer and closure .
382 382 382 383 385 385 386 386 387 388 389 390 391 392 395 395 396 397 399 400
The strength of nonstandard analysis H. Jerome Keisler *
A
Abstract
weak theory nonstandard analysis, witL types at. all finite levels over
both the integers and hyperintegers, is developed
as a
possible framework
for reverse mathematics. In this weak theory we investigate .
of standard part principles and saturation principles which
the strength
are
often used
in practice along with first order reasoning about the hyperintegers to obtain secon d order conclusions about the integers.
1.1
Introduction
In this paper we revisit the work in [5] and [6], where the strength of nonstandard analysis is studied. In those papers it was shown that ·weak fragments of set theory become stronger when one adds saturation principles commonly used in nonstandard analysis. The purpose of this paper is to develop a framework for reverse mathe matics in nonstandard analysis. Vvc ·will introduce a base theory, "weal.v.t of type O" + T from a variable
The application operator which builds a term s of type O" and t of type O" + T.
t( s) of type T from terms
Given terms r, t and a variable v of the appropriate types, r (t j v) denotes the result of substituting t for v in r. Given two terms s, t of type O", s = t will denote the infinite scheme of formulas r ( s/ v ) = r(tjv) where v is a variable of type O" and r ( v ) is an arbitrary term of type N. = is a substitute for the missing equality relations at higher types. The axioms for PRAw are as follows . •
Each axiom of
•
The induction scheme for quantifierfree formulas of L(PRAw).
•
Primitive recursion:
PRA. = m,
=
R(m, f, s(n )) f (n, R(m, j, n)). R(m, j, 0) Cases: c(O, u, v ) = 11, c(s(m). u, v ) = v. Lambda abstraction: (>.u.t)(s) = t(sju) . The order relations < , ::; on type N can be defined in the usual way by •
•
quantifierfree formulas. In [ 1] additional types O" x T, and termbuilding operations for pairing and projections with corresponding axioms were also included in the language, but
6
1 . The strength of nonstandard analysis
as explained in [ 1 ] , these symbols are redundant and are often omitted in the literature. On the other hand, in [ 1] the symbols for primitive recursive functions are not included in the language. These symbols are redundant because they can be defined from the primitive recursive operator R, but they are included here for convenience. We state a conservative extension result from [ 1] , which shows that PRAw is very weak. Proposition 1 PRAw is a co nservative extensio n of PRA, that is, PRAw and PRA have the same co nsequences in L(PRA) . The natural model of PRAw is the full functional superstructure V (N), which is defined as follows. N is the set of natural numbers. Define VN (N) = N , and inductively define Va_,T (N) t o b e the set of all mappings from Va (N) into VT (N). Finally, V (N) = Ua Va (N). The superstructure V (N) becomes a model of PRAw when each of the symbols of L(PRAw) is interpreted in the obvious way indicated by the axioms. In fact. V (N) is a model of much stronger theories than PRAw, since it satisfies full induction and higher order choice and comprehension principles.
1.3
The theory NPRAw
In [1] , Avigad introduced a weak nonstandard counterpart of PRAw, called NPRAw. NPRAw adds to PRAw a new predicate symbol S(·) for the standard integers (and Srclativizcd quantifiers V5 , 3 5), and a constant H for an infinite integer, axioms saying that S(·) is an initial segment not containing H and is closed under each primitive recursive function, and a transfer axiom scheme for universal formulas. In the following sections we will use a weakening of NPRAw as a part of our base theory. In order to make NPRAw fit better with the present paper, we will build the formal language L(NPRAw) with types over a new formal object *N instead of over N. The base type over *N is *N, and if J, T are types over *N then O"+ T is a type over *N. For each type O" over N, let *J be the type over *N built in the same way. For each function symbol u in L(PRAW) from types a to type T, L(NPRAW) has a corresponding function symbol *u from types *(J to type *T . L(NPRAw) also has the equality relation = for the base type *N, and the extra constant symbol H and the standardness predicate symbol S of type *N. We will usc the following conventions throughout this paper. When we write a formula A ( v) , it is understood that vis a tuple of variables that contains
1 .4.
The theory
WNA
7
all the free variables of A. If we want to allow additional free variables we write We will always let:
A(v, . . .) . •
m, n, . . . be variables of type N, x, y, . . . be variables of type *N, j, g, . . . be variables of type N + N. To describe the axioms of NPRAw we introduce the star of a formula of L(PRAw). Given a formula A of L(PRAw), a star of A is a formula *A of L(NPRAw) which is obtained from A by replacing each variable of type J in A by a variable of type *J in a one to one fashion, and replacing each function symbol in A by its star. The order relations on *N will be written l. Nous dirons maintenant qu'un nombre x est un infinitesimal ccrirons X � 0, si
( 1 ')
relatij, et nous (2)
et que
x est un infiniment grand relatif si vimpl > 0 l x l > l
(2')
ce qui s'ecrira x � +oo. (VImpx F(x) est une abreviation pour Vx (Imp(x) =? F(x)) , ce qui s'oralise « pour tout x impropre, F(x) » ) . On peut montrer que tout nombre relativement infinitesimal est infinitesi mal. Un nombre non relativement infiniment grand est dit relativement limite. On fixe ensuite une fois pour toute : un nombre N relativement infiniment grand,  une fonction de Dirac relative a droite de 0, 6.
43
3.4. Calcul de Heaviside sans transformee de Laplace
Cela signifie que 6 reste une fonction de Dirac meme pour un observateur ideal capable de "voir" des fonctions de Dirac impropres. Cette fonction 6 est tres impropre, elle est nulle en dehors d'un intervalle [0, h] avec h � 0. J 'ai defini ensuite dans [15] une collection N de fonctions negligeables de classe c= de deux variables X et t. La generalisation a un nombre plus grand de variables ne pose pas de probleme. On peut meme etendre nos resultats aux cas ou on a un nombre infiniment grand, impropre, de variables. Cela peut etre utile si on a besoin de faire intervenir des parametres qui sont des fonctions standard, representees par un ensemble fini impropre contenant toutes leurs valeurs standard. La classe N vcrifie les proprictcs suivantes. Si a(x, t) E N alors a(x, t ) � 0 pour tout x relativement limite et tout t rclativement appreciable .
t · N c N.
Si * designe le produit de convolution des fonctions, alors
 N * N c N. 1 * N C N pour toute bonne fonction f. an +mN c N, pour taus n E N et m E Z. limites axm atn
On remarque que dans la derniere propriete m peut etre negatif. La derivee d'ordre negatif  k d'une fonction a s'obtient en calculant l'integrale
On integre k fois
Remarque.
N * N, 1 * N ou
La classe N, "n'est pas" un ensemble pas plus que t N, Les inclusions precedentes sont des inclusions de ·
�::';;!: .
collections. On dcfinit alors l'cgalitc de Laplace £: en posant
F
£
=
G
Def ¢'?
::Ja E N F

G
=
!:a
ou !:a est la transformee de Laplace classique de a, qui doit done exister. On pose ensuite
La fonction (J:cl") (p) n'est pas cgalc a 1 , comme pour la distribution de Dirac. On trouve parfois dans les livres de physique, concernant les fonctions de Dirac, l'cgalitc (J:ll) (p) = 1 . En rcalitc pour une fonction de Dirac D. non standard, on a ( 1:6) (p) 1 pour les valeurs limitees de p. Notre fonction 6, fixee plus haut , vcrifie quant a elle rv
3. Analysis of various practices of referring
44
pour tout
p > 0 relativement limite, ( £.6) (p)
�
1.
Dcmontrons ce dcrnier point. La deuxicmc formule de la moyenne donne, pour chaque p > 0,
( L. 6) (p)
=
l
epeh h 6 ( t ) dt, e E [0, 1] .
h � 0 implique Bh � 0, pBh � 0 pour p relativement limite et done, epeh � 1 . Comme J0h 6(t ) dt � 1 , on en deduit que ( L. 6) (p) � 1 pour tout p relative
ment limite.
Notation. On ecrira .f transformation de Laplace.
:::JJ
F si F
£
L.f et .f :::J F si .f admet F comme
Quelques resultats obtenus Pour toutes bonnes fonctions f et g standard ou impropre 1 . Lf est Laplaceindependant de 6. Des choix distincts de la fonction donneraient des transformations Laplaceegales. 2. Si f :::JJ F et g :::JJ F, alors f = g. 3. S i f est de type exponentiel, f :::J F =? f :::JJ F .
4. 5. 6.
Ce qui precede implique que toutes les fonctions qui figurent dans les tables de transformations de Laplace sont des images generalisees. (Lf') (p) � p(Lf) (p)  f(O+) · ( L. 6) (p) . t L f(s) ds (p) � (Lf) (p) .
(i
)
j(t )
diverge.
·
L
£
n =O
aj £ aLj .
8 . L ax
�
£ (Lf) (Lg) . += = an t n alors LF
L(f * g)
7. Si
3.5
6
ax
N
L
n =O
an
+= , mcme si la scrie an n+ L n. l pnn.+ 1 =O p I
I
n
. . etc.
Exemples
Exemple 1 . Recherche d e l a solution d e l'equation aux derivees partielles d€dinies pour X > 0, t > 0. 2 a u (x, t ) ax 2

a u (x, t) = at
e t2
45
References
u ( O+ , t )
=
u ( x, 0 + )
� ( t) ,
=
0 pour tout x.
� est une fonction impropre de Dirac. L' application du calcul formel generalise donne la solution "exacte"
u ( x , t)
=
[� (t)  H (t)
*
et2
J 2� e *
4�2 +
H ( t) et2 • *
Cependant LA REPONSE PHYSIQUE CORRECTE au probleme est la suivante : Pour taus x > 0 et t > 0. (a) Si t est appreciable la solution est indiscernable de
(b) On ne connait pas avec precision le comportement de la solution quand est tres petit . Cette imprecision est heritee du celle de la fonction � .
t
Exemple 2. La fonction e  t2 a une transformation d e Laplace e t admet u n developpe ment en serie de rayon infini, cependant sa transformee de Laplace n ' est pas egale a la somme  qui n'est pas definie  des transformees de Laplace des termes de la serie. On a cependant
Pour la fonction e t 2 il n 'y a pas de transformee de Laplace mais on peut ecrire une image generalisee : t2 "" (2n) !
e :::JJ
6
2 n < txl
n l n+ l "
.p
En conclusion, je dirai que l'examen systematique des questions de seman tique peut accroitre considerablement l'efficacite de l'outil mathematique. Nous venons de le constater pour les mathcmatiques de la physique.
References [1] B . DAMYANOV, "Multiplication of Schwartz distributions and Colombeau Generalized functions", Journal of Appl. Anal . , 5 ( 1 999) , 24960. [2] P . A . l\1 . DIRAC, "The physical interpretation of the Quantum Dynamics", Proc. of the Royal Society, section A, 1 1 3 ( 1 92627) 62 1641. [3] P . A . l\ 1 . DIRAC, Les Gabay, Paris, 1990.
principes de la mecaniq1L e quantique
( 1 93 1 ) , .Jacques
46
3. Analysis of various practices of referring
[4] G . G . G RANGER,
L 'irrationnel,
Editions Odile Jacob, 1 998.
[5] .J . P . G RENIER, "Representation discrete des distributions standard", Osaka J. of Math. , 32 ( 1 995) 7998 1 5 . [6] l\1 . KINOSHITA, "Nonstandard representations of distributions". Osaka .J. Math. , I 25 (1988) 805824 and II 27 ( 1 990) 843861 . [7] H . KoMATSU, "Laplace transform of Hyperfunctions a new foundation of the Heaviside Calculus", .J . Fac. Sci. Tokyo, Sect . lA. Math. , 34 ( 1 987) 805820. [8] G . LUMER and F. NEUBRANDER, The Asymptotic Laplace Transform and evolution equations, Advances in Partial differential equations, Math. Topics Vol. 16. WileyVCR 1999. [9] G . LUMER and F. NEUBRANDER, Asymptotic Laplace Transform and re lation to Komatsu's Laplace Transform of Hyperfunctions, preprint 2000. [10] .J . MIKUSINSKI. Bull. Acad. Pol. Scr. Sci. Math. Astron. Phys. , 43 (1966) 5 1 113. [ 1 1] E. NELSON, " Internal set theory : a new approach to nonstandard analy sis", Bull. A.M . S . , 83 ( 1 977) 1 165 1 1 98. [12] Y . P ERAIRE, "Theorie relative des ensembles internes", Osaka .J. Math. , 29 ( 1 992) 267297. [13] y. P ERAIRE, "Le replacement du referent dans les pratiques de l'analyse issues de Edward Nelson et de Georges Reeb", Archives Henri Poincare, Philosophia Scientiae (2005 ) , a paraitre. [14] Y. PERAIRE, "A mathematical framework for Dirac's calculus", Bulletin of the Belgian Mathematical Society Simon Stevin, (2005 ) , a paraitre. [ 1 5] Y . PERAIRE, Heaviside calculus with no Laplace transform, Integral Transform and Special Functions, a paraitre. [16] C . R A J U , "Product and compositions with the Dirac delta function", J. Phys. A : Marh. Gen . , 1 5 ( 1 982) 38196. [ 1 7] K . D . STROYAN and W .A . .J . LUXEMBURG, Introduction infinitesimals, Academic Press. NewYork, 1976 .
to the theory of
[18] L . ScHWARTZ, "Sur l'irnpossibilite de la multiplication des distributions", C. R. Acad. Sci. Paris, 239 ( 1 954) 847848. [19] L . ScHWARTZ,
Theorie des distributions,
Hermann, Paris ( 1 966)
[20] J. C . VIGNEAUX, "Sugli integrali di Laplace asintotici", Atti Accad. N az. Lincei, Rend. Cl. Sci. Fis. Math. , 6 (29) (1939) 396402.
Stratified analysis? Karel Hrbacck*
It is now over forty years since Abraham Robinson Tealizcd that " the con cepts and methods of Mathematical Logic aTe capable of pmvid·ing a suitable .frnmewoTk .for the development of the D·iffemntial o.nd Jntegml Calc·u.l·us by means of infinitely small and i1Jfin'itely laTge numbeTsn ( Robinson [29] , Intro
duction. p. 2). The mag11itude of Robinson's achievement. cannot be overstated. Not only docs his framework allow rigorous paraphrases of many arguments of Leibuiz, Euler a.nd other mathematicians from the classical period of calcu lus; it has enabled the development of entirely new, important mathematical techniques and constructs not anticipated by the classics. Researcl1ers work ing with the methods of nonstandru·d analysis have discovered new significant results in diverse areas of pure aud applied mathematics, from number theory to mathematical physics and economics. It seems fair to say, however, that acceptance of "nonstandard" methods by t.he lcu·gcr mathematical commnnit;y lags far behind their successes. [n particu lru·, the oftexpressed hope that infinitesimals would now replace the notorious c:t5 mei,hocl in teaching calculus remains umoalized, in spite of notable efi'orts by Keisler [20] , Stroyan [311, I3enci and Di Nasso [4] , and others. Sociological reasons  the inherent conscrvativity of the mathematical community, t.he lack of a concentrated effort at proselytizing  are often mentioned as an explana tion. There is also the fact that "nonstandard" methods, at least in the form in which they are usually presented, require heavier reliru1ce on formal logic thru1 is customary in mathematics at large. While acknowledging much truth to all of the above, here I shall concentrate on another contributing difficulty. At the risk of an overstatement, it is this: while it is undoubtedly possible to do calculus by means of infinitcsimals in the Robinsonian framework, it does not seem possible to do calculus only by means of infinitesimals in it. In particular, the promise to replace the c:6 method by the use of infinitesimals cannot be carried out in full. *
Department o f 1'vlathcmat.ics, The Chy College o f CUNY, New York, NY 1 0031 .
[email protected] . cuny . edu
4. Stratified analysis?
48
In Section 4 . 1 I examine this shortcoming in detail, review earlier relevant work, and propose a general plan for extending the Robinsonian framework with the goal of remedying this problem and  possibly  diminishing the need for formal logic as well. Section 4.2 contains a few examples intended to illustrate how mathematical arguments can be conducted in this extended framework. Section 4.3 presents an axiomatic system in which the techniques of Section 4.2 can be formalized, and discusses the motivation and prospects for its further extension.
4. 1
The Robinsonian framework
Here and in the rest of the paper, by Robinsonian framework I mean any presentation of "nonstandard" methods that postulates a fixed hierarchy of standard, internal and, in most cases, also external sets. Thus the origi nal typetheoretic foundations of Robinson [29] , the superstructure method of Robinson and Zakon [30] ( also Chang and Keisler [6] ) , and direct use of ul trafilters a la Luxemburg [21] , as well as axiomatic nonstandard set theories like HST [13, 18, 19] or Nelson's 1ST [23] , are covered by the term, and the dis cussion in this section applies to all of them. I present the arguments in the "internal picture" employed in 1ST and HST; that is, for example, R denotes the set of all ( internal ) real numbers, and is referred to as the standard set of reals ; if needed, 0R denotes the external set of all standard reals. Superstruc ture afficionados would use *R for R and R for 0R. The same conventions apply to the standard set of natural numbers N and other standard sets. The paradigmatic: example below, the familiar nonstandaTd definition of continuity, illustrates the difficulty I am concerned about . Definition 4. 1 . 1
Let f
:
R + R
be a standard function, and x
ER
a stan
dard real number. (i) f is continuous at x iff for all infinitesimal h, f ( x + h)  f ( x) is infinitesimal. (ii) f is (pointwise) continuous iff for all standard x E R, f is continuous at x. Explicitly: (\istx E R) (\i infinitesimal h) (! ( x+h) f ( x) is infinitesimal) .
It is a basic and useful fact of nonstandard analysis that the notion of continuity of a standard function at a standard point defined above can be extended in a natural way to the notion of continuity at a nonstandard point, and that a standard continuous function f : R + R is continuous at all x E R, even the nonstandard ones. But, what precisely does this mean in the Robinsonian framework, and how do we know that it is true? Certainly not by transfer! In the Robinsonian framework, for a statement about standard
49
4. 1 . The Robinsonian framework
obj ects to be transferable from the standard to the internal universe, all of its quantifiers have to range over standard sets; formally, it has to be of the form yst where y is an Eformula ( internal formula) . Definition 4. 1 . 1 is not of this form; the quantifier (V infinitesimal h) ranges over internal sets. Briefly, Definition 4 . 1 . 1 is not transferable. A naive attempt to transfer 4. 1 . 1 ( ii ) will likely produce something along the lines of
(Vx E R) (V infinitesimal h) (f( x + h)  f( x) is infinitesimal ) . As is well known. this statement is equivalent to ·uniform continuity of f ( for standard f) . How then do we arrive at our "basic and useful fact"? Every treatment of elementary nonstandard analysis has to answer this question somehow. More over, similar difficulties appear with derivatives, integrals in fact, with every concept defined by nonstandard methods. There seems to be little explicit attention paid to this issue in the literature. Important exceptions are the writings of Peraire [25][27] , Gordon [ 1 1 1 2] , and Andreev's thesis [ 1 ] ; dis cussions of a number of points considered in this paper can be found there. I realized the crucial importance of this issue for teaching of nonstandard anal ysis during O'Donovan's talk in Aveiro. While describing his experiences with the nonstandard definition of derivative, O'Donovan recounted some questions his students typically ask: "Can we use this formula when x is not standard? When f is not standard?" The answer of course is NO but what then are they supposed to use? After alL a standard function like sin x does have a derivative at all x!1 Three implicit responses applicable i n Robinsonian framework can b e dis cerned. ,
Response I ( Robinson [29] , Goldblatt [ 1 0] ) . Although Definition 4. 1 . 1 is not expressible by an Eformula, it is to an Eformula, namely, to the standard definition of continuity.
equivalent
Definition 4 . 1 .2 Let f : R + R be a standard function, and x E R standard. f is continuous at x iff ( V8t c > 0) (38tcl > O) (V8t y E R) ( I Y  x i < c5 =?
l f(y )  f( x) l < c ) .
The formula on the right side of Definition 4 . 1 . 2 is transferable, and yields a natural notion of continuity for all f : R + R and all x E R that agrees with Definition 4. 1 . 1 for standard f and x. 1 I a m grateful t o
R.
O 'Donovan for many subsequent email exchanges that have been
extremely helpful in further clarification of the difficulties with using infinitesimals to teach calculus.
50
4. Stratified analysis?
Definition 4 . 1 . 3 Let f : R + R real. f is continuous at x iff
(Vc > 0) (3 el
>
be any (internal) function, and x E R any
O ) (Vy E R) ( I Y  x i < el ==;. i f(y)  f(x) i < c ).
The problem i s resolved, but at the cost o f a relapse t o the usual Eel definition of continuity, at the internal level. This response also illustrates one of the chief reasons why nonstandard methods have to rely so heavily on formal logic. In Definitions 4. 1 . l (i) and 4 . 1 . 2 we have two equivalent formulas, of which one is transferable and the other is not . Transferability is an attribute of formulas; it is a logical, metamathematical concept. Response II (Nelson [23] ) . First, we use Definition 4. 1 . l (i) to define continuity for standard j, x . Then we let C := s { (!, x) : f : R + R, x E R, j, x st andard, f continuous at x} (C is a standard set ) , and finally, for any (internal) f and x, define Definition 4. 1 .4 f
is
continuous at x
iff (! , x) E C.
If f is standard and continuous at all standard x E R then (vst x) ( (f. x) E C) , this formula is transferable, and gives (Vx) ( (!, x) E C ) , i.e . , f is continuous at all x E R, as desired. The problem here is that this ("somewhat implicit" [23] ) definition of con tinuity is completely divorced from the usual intuition (captured for standard r X, in different ways, both by the standard Eel definition and by the non standard definition using infinitesimals) : a function f is continuous at x if arguments "ncar" x yield values ..ncar" f(x). According to 4.1 .4, the im plicit meaning of the statement "f is continuous at x" for standard f and unlimited x is "x E 8{y E R : y standard, f continuous at y } " ; it is not re lated to the behavior of f near x in the sense of the order topology on R . Similarly, continuity of f(x) : = xv , where v E N i s unlimited, translates t o " v E 8{n E N : f (x) : = xn i s continuous }"; i.e., xv i s continuous "because" n xn is a continuous function for all finite n! Definition 4 . 1 . 4 can be decoded via Nelson's reduction algorithm; it then gives the usual Eel definition, at the internal level, as in Response I. We note that here too there are two equivalent formulas, one of which transfers and the other does not. Response III. This is a feasible response in an approach based on superstructures, even though I found no discussion of it in the literature. While considering it I switch to the asterisk notation.
51
4. 1 . The Robinsonian framework
We recall that the definition of (pointwise) continuity in terms of infinitesi mals can be generalized to standard f : T1 + T2 , where T1 and T2 are arbitrary standard topological spaces. The internal open sets of * R form a base for an (external) topology on * R , the Qtopology. The meaning of "f is continuous at x " for internal f and x E * R can be given by "f is continuous at x in the Qtopology on * R . " As noted above, this concept has a nonstandard def inition, although applying it requires working with 0 ( * R ) in a "secondorder" cnlargemcnt 2 of the superstructure that contains * R . Yet the difficulty is not resolved. In order to prove that the two definitions of continuity are equivalent for standard f and x, one needs to apply transfer to the (equivalent) standard definitions in terms of open neighborhoods, i.e., fall back on the c;6 method, in topological disguise. The Robinsonian framework does not provide for direct transfer of nonstandard definitions. This response also begs the question, how do we know that ® ( * f ) is continuous, and so on. Clearly, an infinite sequence of consecutive enlargements would be needed. It seems that every attempt to define continuity ultimately has to be grounded on the E6 method. As remarked above, the same difficulty appears with derivatives, integrals in fact, with all standard concepts introduced by nonstandard methods. I sec it as a serious problem for the Robinsonian frame work, if not as a research tool, surely as a teaching tool and, fundamentally, as a satisfactory answer to the question about the place of infinitcsimals, and nonstandard objects in general, in mathematics. What is to be done? Contemplation of the three responses suggests some ideas. First, we need to abandon the fixed distinction between standard and nonstandard and be able to treat any (internal) object as if it were standard, and in this capacity subject to application of nonstandard definitions and theorems. This is the idea of relativization of standardness. Second, we need to be able to transfer properties described by arbitrary (external) formulas, not just Eformulas; for emphasis, I refer to this facility as general transfer. Both ideas have some history in the literature of nonstandard analysis. A definition of relative standardness seems to appear first in Cherlin and Hirschfeld [7] , although its modeltheoretic roots can be discerned in [6] ; but the subsequent development occurred mostly in the axiomatic setting. Gor don [ 1 1] defined two notions of relative standardness in 1ST (one of them is essentially the same as in [7] ) . Wallet [24] proposed to use a binary relative 2
For some applications of "secondorder" enlargements see Molchanov
[22] .
In the alpha
the * embedding is defined for all sets, but * * w1saturated, and monads in the Qtopology on * are trivial. theory of Benci and DiNasso
[4] .
R
(
R) is only
52
4.
Stratified analysis?
standardness predicate as a primitive in an axiomatic treatment, an idea that was developed systematically by Peraire in [25] . The notion of relative stan dardness stratifies the universe into levels of standardness. Both Gordon and Peraire give a nonstandard definition of continuity applicable to all f and x (see Definition 4.2.2), and numerous other examples (see also [ 1 , 12, 26, 27, 28] ) . Gordon's approach docs not work as smoothly for concepts whose definitions involve shadows, such as derivative , because standardization does not hold in full. A sufficiently strong (for this purpose) standardization docs hold in Peraire's theory RIST. Another difference is that, unlike Gordon's, Peraire's relative standardness predicate is a total preordering. Yet another stratified nonstandard set theory (not employing a binary relative standardness predi cate) was put forward by Fletcher [9] . None of [9, 11, 25] give an explicit formulation of transfer for more than just Eformulas. To my knowledge, the idea of (more) general transfer appears first in the work of Benninghofen and Richter [5] . The main result of [5] is a transfer theorem for a certain (complicated) class of Estformulas; Cutland [8] gives some simpler special cases. Although the class of transferable formulas is limited, it has led to interesting applications (see the proof of l'Hopital rule in Section 4.2, and [5, 32 , 33] ) . However, the idea of relative standardness is not explicit in [5] . Further discussion of the mutual relationship of these various approaches can be found in [16] . In my opinion, the decisive step needed to resolve the difficulty discussed above is to combine relative standardncss with general transfer. This step was taken by Peraire in [26] with his proof (in RIST) of "polytransfer," essentially, transfer for all formulas that do not quantify over levels of standardncss. Many standard concepts have satisfactory nonstandard, transferable definitions in RIST. Nevertheless, there are situations (see Example 4 in Section 4.2) where quantification over levels of standardness is both natural and necessary. More over, the need to single out the special classes of formulas to which principles of RIST apply increases reliance of the framework on formal logic. It is my belief that for a theory of the "nonstandard " to be fully satis factory, both foundationally and practically, all of its principles need to apply uniformly at all levels, and to all formulas. In addition to a complete resolution of the difficulty that is the subject of this discussion, such framework would also diminish the need for appeals to formal logic in practical work: all for mulas would be transferable, and equivalent formulas would have equivalent transfers. The axiomatic system FRIST presented in Section 4.3 achieves these objectives for internal sets. The examples in the next section show some of the power of internal methods extended by relativizcd standardncss and general transfer.
4.2.
53
Stratified analysis
4.2
Stratified analysis
This section gives several examples intended to illustrate "nonstandard" mathematics in stratified framework. The presentation is informal; an ax iomatic system FRIST in which all of the arguments in this section can be formalized is described fully in Section 4.3. Our basic assumption is that the universe of mathematical objects (= sets) is stratified by a binary relation � into "levels of standardness." The notation x � y is to be read "x is standard relative to y" or "x is ystandard", and it is a dense total preordering with a least element 0. The 0standard sets are called simply standard; they form the lowest level of standardness. The class of xstandard sets is denoted §x . Let a be a set; relativization of standardness to level a consists in regarding §a , rather than §o , as the lowest level of standardness. A statement about standard sets is relativized to level a by replacing all references to "standard" with " astandard" (more explicitly, by replacing "x is ystandard" with "x is
(y, a)standard") .
The key principle that governs the stratified universe is general transfer:
All valid statements about standard sets remain valid when relativized to any level a. Mathematical practice proceeds by enriching the language with new defini tions. We make it a general rule that, whenever some standard notion (a new predicate or function) is defined for standard x in terms of some property of x, the definition of the notion is extended to all x by rclativizing the defining statement to level x (the definition has to be fully relativized see Section 4.3) . In addition, we make the familiar assumptions that •
•
•
all the usual mathematical operations preserve standardness §o of all standard sets satisfies ZFC) ;
( the class
given any property of x and any standard set A, there is a standard set B, whose standard elements are precisely those standard x E A having that property (standardization) : for every level a :::J 0 there exist astandard unlimited natural numbers (a much stronger idealization is available  see Section 4.3  but this suffices for calculus) .
The examples that follow illustrate the formulation of relativizations and the general rule. Definition 4.2. 1
(i) r E R is alimited iff l r l
4. Stratified analysis?
62
[7] G . CHERLIN and J . HIRSCHFELD, Ultrafilters and ultraproducts in non standard analysis, in Contributions to Nonstandard Analysis, ed. by W.A.J. Luxemburg and A. Robinson, North Holland, Amsterdam 1972. [8] N . J CUTLAND, "Transfer theorems for Jrmonads". Ann. Pure Appl. Logic, 44 ( 1989) 5362. .
.
[9] P. FLETCHER, "Nonstandard set theory", J. Symbolic Logic, 54 ( 1989) 10001008. [10] R. GoLDBLATT, Lectures on the Hyperreals: An Introduction to Nonstan dard Analysis, Graduate Texts in Math. 188. SpringerVerlag, New York, 1998. [ 1 1] E.I. GORDON, "Relatively nonstandard elements in the theory of inter nal sets of E. Nelson", Siberian Mathematical Journal, 30 (1989) 8995 ( Russian ) . [12] E.I. GORDON, Nonstandard Methods in Commutative Harmonic Analysis, American Mathematical Society, Providence, Rhode Island, 1997. [13] K. HRBACEK , ·· Axiomatic foundations for nonstandard analysis·· , Funda menta Mathematicae, 98 (1978) 1  19; abstract in .J. Symbolic Logic, 4 1 (1976) 285. [14] K . HRBACEK, "Nonstandard set theory", Amer. Math. Monthly, 86 ( 1979) 1  19. [15] K. HRBACEK , Internally iterated ultrapowers, in Nonstandard Models of Arithmetic and Set Theory, ed. by A. Enayat and R. Kossak, Contempo rary Math. 36 1, American Mathematical Society, Providence, R.I.. 2004. [16] K . HRBACEK, Nonstandard objects in set theory. in Nonstandard Methods and Applications in Mathematics, ed. by N.J. Cutland, M. Di Nasso and D .A. Ross, Lecture Notes in Logic 25, Association for Symbolic Logic, Pasadena, CA. , 2005, 41 pp. [17] K . HRBACEK , Relative Set Theory, work in progress. [18] V. KANOVEI and l\ I . REEKEN, "Internal approach to external sets and universes", Studia Logica, Part I, 5 5 (1995) 227235; Part II, 55 ( 1995) 347376: Part III, 56 (1996) 293322. [19] V. KANOVEI and M. REEKEN, Nonstandard Analysis, Axiomatically, SpringerVerlag, Berlin, Heidelberg, New York, 2004.
References
63
[20] H . J . KEISLER, Calculus: An Infinitesimal Approach, Prindle, Weber and Scmidt, 1976, 1986. ,
[21] W . A . .J . L UXEMBURG A general theory of monads, in: W.A.J. Luxem burg, ed., Applications of Model Theory to Algebra, Analysis and Proba bility, Holt, Rinehart and Winston 1969. [22] V.A. M oLCHANOV, "On applications of double nonstandard enlargements to topology", Sibirsk. l\Iat. Zh. , 30 (1989) 6471. [23] E. NELSON, "Internal set theory: a new approach to Nonstandard Anal ysis", Bull. Amer. Math. Soc., 83 ( 1977) 1 165 1 198. [24] Y. PERAIRE and G. WALLET, "Une theorie relative des ensembles in ternes", C.R. Acad. Sci. Paris, Ser. I, 308 (1989) 301 304. [25] Y . P ERAIRE, "Theorie relative des ensembles internes", Osaka Journ. Math., 29 ( 1992) 267297. [26] Y. PERAIRE, "Some extensions of the principles of idealization transfer and choice in the relative internal set theory", Arch. Math. Logic, 34 (1995) 269277. [27] Y. P ERAIRE, "Formules absolues dans la theorie relative des ensembles internes", Rivista di l\Iatematica Pura ed Applicata, 19 (1996) 2756. [28] Y. P ERAIRE, "Infinitesimal approach of almostautomorphic functions", Annals of Pure and Applied Logic, 63 ( 1993) 283297. [29] A. RoBINSON, Nonstandard Analysis, Studies in Logic and the Founda tions of Mathematics, NorthHolland. Amsterdam, 1966. [30] A. RoBINSON and E. ZAKON, A settheoretical characterization of en largements, in W.A . .J. Luxemburg, ed . . Apphcations of Model Theory to Algebra, Analysis and Probability, Holt, Rinehart and Winston 1969. [31] K . D . STROYAN, Foundations of Infinitesimal Calculus, 2nd ed. , Academic Press 1997. [32] K . D . STROYAN, B . BENNINGHOFEN and M . l\1 . RICHTER, "Superin finitesimals in topology and functional analysis". Proc. London Math. Soc., 5 9 ( 1989) 153181. [33] K . D . STROYAN , Superinfinitesimals and inductive limits, in Nonstandard Analysis and its Applications, ed. by N. Cutland, Cambridge University Press, New York, 1988.
ERNA at work * ** C. Impens and S. Sanders
Abstract
Elementary Recursive N on standard Analysis . i n short ER.NA , is a con structive system of nonstandard analysis proposed around 1995 by Chuaqui, Suppes and Sommer. It has been shown to be con sistent and. without stamlard part ftmctiou or continuum, it allows major parts of analysis to be develop ed in an applicable form. vVe briefly discuss ER.N A's foundations and use them to prove a supremum principl e and provide a square root fuuctiou, b oth up t;o iufinit;esimals.
5.1
Introduction
Hilber·t 's Program, proposed in 1921, called for an axiomatic formalization of mathematics, together with a proof that this axiomatization is consistent. The consistency proof itself was to be carried out using only what Hilbert called finitary methods. The special chamct:er of finitru·y reasoning then would justify classical mathematics. In due time, many chmadcrized Hilbert's infor mal notion of '.finitruy' as that which can be formalized in Primitive Recursive Arithmetic (PRA) , proposed in 1923 by Skolem. In PR.A one finds (a) an absence of explicit quantification, (b) an ability to define primitive recursive functions, (c) a few rules for handling equality, e.g. , substitution of equals for equals, (d) a rnle of instantiation, and ( e) a simple induction principle.
Godel's second incompleteness theorem (1931) it became evident that 7Jar·tial realizations of Hilbert's prognun are possi ble. The system pro posed by Chuaqui and Suppes is sueh a partial realization, in that it pro vide� ru1 a.xiomatic foundation for basic analysis, with a PRA consistency proof ([1], p. 123 and p. 130) . Sommer and Suppes's improved system al lows definition by recursion (which does away with a lot of explicit axioms) By
only
•Department of Pure Mathcmatk'3 and Computer Algebra, Universi l.y of Gent, Delgiw11. ··
[email protected] . ugent . ac .be
[email protected] . ugent . be
5.2. The system
65
and still has a PRA proof of consistency ([2] , p. 21 ) . This system is called Elementary Recursive Nonstandard Analysis, in short ERNA. Its consistency is proved via Herbrand's Theorem ( 1930) , which is restricted to quantifierfree formulas Q(x1 , . . . , xn ) , usually containing free variables. Alternatively, one might say it is restricted to universal sentences (\ix l ) . . . (\ixn )Q(x l , . . . , Xn ) ,
obtained by closing the open quantifierfree formulas by means of universal quantifiers. Herbrand's theorem states that, if a collection of such formulas resp. sentences is consistent, it has a simple 'Herbrand' model and, if it is not, its inconsistency will show up in some finite procedure. Herbrand's theorem requires that ERNA's axioms be written in a quantifier free form. As a result, some axioms definitely look artificial; fortunately, the orems don't suffer from the quantifierfree restriction. Calculus applications of ERNA have been, so far, scarce and sketchy. Thus, [3] contains an outline of an existence theorem for firstorder ordinary differential equations, relying on the property, stated without proof, that a continuous function on a compact interval is bounded. As part of a less anec dotical approach we will provide an ERNA version of the supremum principle and deduce from it a square root function. Both results hold up to infinitesi mals; as ERNA has no standard part function, it is intrinsically impossible to do better. 5.2
The system
The system we are about to describe was first presented in [2] , and all our undocumented results are quoted from that paper. The foundations are also exposed, in a more informal manner, in [ 3] . Notation 5 . 2 . 1 N
=
{ 0, 1 , 2, . . . } consists of the (finite) integers.
Notation 5.2.2 x stands for some finite
(possibly empty) sequence (x 1 , . . . , xk ) .
denotes a term in which x = (x 1 . . . . , xk) is the list of the distinct free variables.
Notation 5.2.3 T(x)
5. 2 . 1
The language
•
connectives:
•
quantifiers: \i , 3
/\,
'
, V, + , +
5. ERNA at work
66 • •
an infinite set of variables relation symbols: 1 =
binary x y binary x � y unary I(x) , read as 'x is infinitesimal', also written 'x � 0 ' unary N(x) , read as 'x is hypernatural ' . •
individual constant symbols:
0 1 c (The Axiom 3 (6) of 5.2.2 shows that c denotes a positive infinites
imal.)  w (The axioms 3 (7) and 2 (4) of 5.2.2 show that w = 1/c denotes an infinite hypernatural.)  I , read as 'undefined'. Notation 5.2.4
1;o =n •
'x is defined' stands for 'x /= I '. (Examples: 1/0 is undefined,
function symbols: 2 (unary) 'absolute value' lxl, 'ceiling ' Ixl , 'weight ' llxll · (For the meaning of llxll , sec Theorem 5.2.3.) (binary) x + y , x  y, x · y, xjy, xAy. (Axiom set 6 and Axiom 12 ( 4) of 5.2.2 show that x A n = xn for hypernatural n , else undefined.) for each k E N, k kary function symbols Kk.i (i 1 , . . . , k) . (The Axiom schema 7 of 5.2.2 shows that Kk, i (x) are the projections of the ktuple x. ) for each formula rp with m + 1 free variables, without quantifiers or terms involving min. an mary function symbol minp. (For the meaning of which, see Theorem 5.2.6 and Theorem 5.2.7.) for each triple (k, a(x 1 , . . . , Xm), T(x l , . . . , Xm + 2 )) with 0 < k E N, a and T terms not involving min, an (m+ 1 )ary function symbol rec�T ' (Axiom schema 9 of 5.2.2 shows that this is the term obtained from a and T by recursion, after the model f(O, x) a(x) , f(n + 1 , x) T(j(n, x), n , x) , if terms are defined and don't weigh too much.) =
=
1 2
For better readibility we expre�s the relations in We denote the values as computed in
x
or
(x, y)
x or
in
(x, y).
according to arity.
according to the arity.
=
5.2. The system 5.2.2
67
The axioms
Axiom set 1
(Logic) . Axioms of firstorder logic.
Axiom set 2
(Hypernaturals).
1.
0 is hypeTTwtuml; 2. if x is hypernatuml, so is x + 1; 3 . if x is hypernatuml, then x � 0; 4  w is hypernatural. 'x is infinite ' stands for 'x /= 0 1\ 1/x � 0 '; 'x is fi nite ' stands for 'x is not infinite '; 'x is natural ' stands for 'x is hypernatural and finite '.
Definition 5.2.1
Axiom set 3
(Infinitesimals) .
1.
if x and y are infinitesimal, so is x + y; 2 . if x is infinitesimal and y is finite, xy is infinitesimal; 3. an infinitesimal is finite; 4  if x is infinitesimal and I Y I :::; x, then y is infinitesimal; 5. if x and y are finite, so is x + y; 6. is infinitesimal; 7. 1/ w . c;
c =
(Ordered field) . Axioms expressing that the elements, with I excluded, constitute an ordered field of characteristic zero with absolute value function. These include (quantifierfree)
Axiom set 4
•
• •
=
=
if x is defined, then x + 0 0 + x x; if x is defined, then x + ( 0  x ) = ( 0  x ) + x = 0; if x is defined and x /= 0, then x (1/x) (1/x) x ·
=
·
=
1.
(Archimedean). If x is defined, Ixl is a hypernatural and 1 < x :::.: lxl x l l
Axiom set 5
Theorem 5 . 2 . 1
If x is defined, then Ix l is the least hypernatural � x.
Theorem 5.2.2
x is finite iff there is a natural n such that l x l
:::;
n.
5. ERNA at work
68 =
Proof. The statement is trivial for x 0. If x /= 0 is finite, so is lxl because, assuming the opposite, 1/ l x l would be infinitesimal and so would 1/x be by axiom ( 4) of set 3. By axiom (5) of the same set, the hypernatural llxll < l xl + 1 is then also finite. Conversely, let n be natural and lxl :::; n. If 1/ l x l were infinitesimal, so would 1/n be by axiom (4) of set 3 , and this contradicts the assumption that n is finite. D Corollary 5 . 2 . 1 Axiom set 6 1.
2.
x � 0 iff l x l < 1/n for all natural n ? 1 .
(Power) .
if x /= I, then xA O = 1 ; if x /= I and n is hypernatural, then xA( n + 1) = (x n) x . A
Axiom schema 7
·
(Projection) .
If x 1 , . . . , Xn are defined, then 1fn , � (x l , . . . , Xn ) = X; for i = 1 , . . . , n. Axiom set 8 1. 2. 3.
(Weight) .
If ll x ll is defined, then ll x ll is a nonzero hypernatural. If l x l = m/n :::; 1 (m and n /= 0 hypernaturals), then ll x ll is defined, ll xll  lxl is hypernatural and ll x ll :::; n. If lxl = m/n ? 1 (m and n /= 0 hypenwturals), then ll xll is defined, llxll/lxl is hypernatural and ll x ll :S m .
Definition 5.2.2
hypernatural.
A hyperrational is of the form ±pjq, with p and q /= 0
Theorem 5.2.3 1. 2.
If x is not a hyperrational, then llxll =1. If x is a hyperrational, say x = ±p/ q with p and q /= 0 relatively prime hypernat11rals, then II ± p/qll
=
max{ I P I , l q l }.
Remark. In both statements of this theorem, the antecedent can be expressed in a quantifierfree way, but the whole sentence cannot. (This explains why it is a theorem and not part of the axioms.) For instance, N(p) + .N(plxl) expresses 'x is not hyperrational'. Theorem 5.2.4 1 . II O II
=
1;
5. 2 . The system 2. 3.
4
69 =
if n ? 1 is hypernatural, li n II n; if llxll is defined, then 11 1/x ll ll x ll and ll lxl l l l lxl l S ll x l l ; if ll x ll and II Y II defined, ll x + Y ll , ll x  Y ll , ll xy ll and ll x/y l l are at most equal to (1 + ll x ll ) ( 1 + II Y II) , and ll xAy ll is at most (1 + ll x ll t ( l + II Y II ) . =
=
Notation 5.2.5
For any 0 < n E N we write
Notation 5.2.6
For any 0 < n E N we write
2's Theorem 5 . 2 . 5 If T(x) is a term not involving w, exists a 0 < k E N such that n
Axiom schema 9
c:,
rec or min, then there
(Recursion) For 0 < k E N, a and T not involving min:
if this is defined, and has weight ::; 2 �:1'11 , if a (x) =I, otherwise. rec�An + 1 , x)
=
{
T(rec�An, x) , n, x) if defined, with weight ::; 2 �x, n+ l ll , if T(rec�T (n. X) , n, X) = I , 1
0
otherwise.
If a is constant , the list x is empty, and the weight requirements mentioned in this axiom schema are void. A few words concerning the restrictions included in this axiom schema. One of ERNA 's main advantages over the ChuaquiSuppes system is, that it allows some form of recursion while preserving a finitary consistency proof. In achieving this, a crucial role is played by the weight function, introduced axiomatically but given explicitly in theorem 5.2.3. Recursion is an essential feature of PRA, and it is therefore impossible to prove inside PRA the con sistency of a system that has unrestricted recursion. ERNA's axiom schema 9 restricts recursion by truncating objects outgrowing the preset weight stan dard. In view of the huge bounds allowed, it seems unlikely that access to calculus applications will suffer from this restriction; computing weights is the price to be paid in practice.
5. ERNA at work
70
(Internal minimum) . For any quantifierfree formula rp( y , x) not involving min or I we have
Axiom schema 10 1.
min�' ( x) is a hypematural number; 2. if minP (x) > 0, then rp(minP (x) , x) ; 3. if n is a hypematural and 'P ( x) ) then n,
min(£) ::; n and rp(min(x) , x) . P P Theorem 5.2.6 If the quantifierfree formula rp( y x) does not involve I or min, and if there are hypematural n 's such that rp( n , x) , then minP (x) is the least of these. If there are none, minP (x) 0. ,
=
Corollary 5.2.2
Proofs by hypematural induction.
Example 5 .2 . 1
The sum of two hypematurals is a hypematural.
Fix any hypernatural x. If the theorem is wrong, there exists at least one y with N(y) /v.N(x + y). By Theorem 5.2.6, there is a least number with these properties, say yo. Then yo /= 0 since x + 0 x (field axiom) and N( x) (assumption) . From yo /= 0 , N(yo  1) (hypernatural axiom) . By leastness, N(x + (yo  1)). Hence (field axiom) N((x + yo)  1) and finally N(x + yo) (hypernatural axiom) . This contradiction proves the theorem. D Proof.
=
(External minimum) . For any quantifierfree formula rp(y, x) not involving min, or E We have 1 . min 'I' ( x) is a hypematural number; 2. if min"' (x) > 0, then rp(min"' (x) . x); 3. if n is a natural numbeT, ll x ll is finite and rp( n , x) , then min'l'(x) ::; n and rp(min'l' (x), x) .
Axiom schema 1 1
Remark. I is
W
allowed in rp.
Let rp( y , x) a quantifierfree formula not involving min, w or If ll x ll is fimte and if there are natural n 's such that rp( n , x), then min'l' (x) is the least of these. If there are none, min'l' (x) 0.
Theorem 5.2.7
c.
=
Corollary 5.2.3
Proofs by natural induction.
Axiom schema 12 1.
0, 1,
w,
c
( (Un)defined terms) .
are defined;
5.2. The system
71
lxl, l x l , llx ll are defined iff x is; 3. x + y, x  y, xy are defined iff x and y are; x / y is defined iff x and y are and y /= 0 ; 4. x A y is defined iff x and y are and y is hypernatural; 5. Kk � ( x l . . . . , x k ) is defined iff x1 , . . . , Xk are; , 6. if x is not a hypernatural, rec�A x, ff) is undefined; 7. min 'P (x1 , . . . , x k ) is defined iff x1 , . . . , Xk are. 2.
Theorem 5 . 2 . 8 (Hypernatural induction)
formula not involving min or I, such that
Let y(x) be a quantifierfree
1.
y(O) holds,
2.
the implication (N(n) 1\ y(n)) + y(n + 1) holds.
Then y(n) holds for all hypernatural n. Suppose, on the contrary, that there is a hypernatural n such that .y(n) . By Theorem 5.2.6, there is a least hypernatural no such that .y(no). By our assumption (1) , no > 0. Consequently; y(no  1) does hold. But then , by our assumption (2) , so would y(no). This contradiction proves the theorem. D Proof.
Example 5.2.2
The sum of two hypernaturals is a hypernatural.
Proof. Fix any hypernatural N and consider the formula N(N + x). Both N(N + 0) and N(N + n) + N(N + n + 1) arc included in axiom set 5.2.2. Hence N(N + n) for every hypernatural n. D Example 5.2.3 Let y(n) be If no < n1 are hypernatu.rals then y(no) = y(n l ) .
a quantifierfree formula not involving min or I . such that no ::; n ::; n1  1 + y(n) y(n + 1), =
The formula y(no) = y(no + x ) holds for x = 0. If y(no) = y(no + n) for any hypernatural no + n ::; n1  1, then also y(no) y(no + n + 1) by D assumption. Hence y(no) = y(no + n) for 0 ::; n ::; n1  no. Proof.
=
For further usc we collect here some definable functions, being terms of the language that ( provably in ERNA) have the properties of the function.
1. The identity function id(x)
=
x is 1f l , l ·
72
5.
ERNA at work
2. For each closed term T and each arity k, the constant function
3. The hypersequence r(n) is rec�T with k =
L
•
{�
=
if n 0 if n ::::> 1
a = 0 , T = c2, 1 ·
4. The function
if X = 0 otherwise
is 1 + x  r ( I I x l l ) . 5.
The functions
h(x) are
{�
x + l x l an d 1 + 2 2( (x)
if X > 0 otherwise
and H(x)
( ( l x l) , respect1ve . · 1y. 2( (x )
6. The function
{�
if X 2;, 0 otherwise
if a < x :::; b otherwise
is h(x  a)H(b  x) . Likewise for the characteristic function of any other interval. 7.
For constants a, b and terms p, a, T, the function
d(x)
=
{
a(x) T(x)
if a < x :::; b and p(x) > 0 otherwise
is 1 ( a . b] (x) (h(p(x)) a(x) + ( 1  h(p(x))) T(x) ) . Likewise for any other type of interval in a < x :::; b and /or any other inequality in p(x) > 0. Any such construction will be called a definition by cases. If no interval is specified, the terms p, a, T and the resulting function can have more than one free variable. The next theorem is to be considered as an ERNA version of the supremum principle for a set of type {x I f(x) < 0}.
5.2. The system Notation 5.2.7
73
We write a « b if a < b and a � b.
Let b < c be constants such that d : = c  b is finite. Further, let f ( x) be a term not involving I or min, such that f ( x) is never undefined for b 5:. x 5:. c. If f(c) ? 0, f(b) < O, then there is a constant 1 with the following properties: f(r) ? 0, iv. for every natural number n ? 1 there are x > 1  1/ n such that f(x) < 0. If f ( x) has the extra property Theorem 5.2.9
z.
zz.
zzz.
(f(x) < 0 1\ b < y < x)
�
f(y) < 0
(5. 1)
then 1 is, up to infinitesimals, the only constant > b with the properties (iii) and (iv). In order to apply recursion, we choose c as our term a and use definition by cases to obtain the term
Proof.
T (t ' n)
=
{
t  dj2 n t
if f(t  dj2n ) ? 0 otherwise.
Note that 'otherwise' is equivalent here to 'if f(t  dj2n ) < 0 ' , because we have excluded undefined values for f(x) . ERNA ' s unary function symbol rec�T for this particular a and T will be shortened to rec. Its properties can be stated simply as rec ( O ) c and rec ( n + 1) T ( rec ( n), n ) because undefined terms cannot occur, and there are no weight requirements because T has arity two. If we prove that for any hypernatural n the two properties =
=
f ( rec ( n )) ? 0 f(rec(n )  d/2n  l ) < 0 ( n ? 1) hold, we are done. It suffices to take
d/2w  l � 0,
1
1 =
(5.2) (5.3)
rec ( w) and to note that, because
rec(w )   < rec ( w)  d/2w  l n
5. ERNA at work
74
for any natural number n ? 1 . We prove ( 5 . 2 ) by hypernatural induction. For n = 0 the requirement ( 5 . 2 ) is identical with the assumption (i) . Now let n be a hypernatural for which ( 5 . 2 ) holds. If f(rec(n)  d/2n ) ? 0, the definition of T implies that rec(n + 1) T(rec(n) , n) rec(n)  d/2 n , which translates the assumption into f(rec(n + 1)) ? 0. Otherwise, rec(n + 1) = T(rec(n) , n) = rec(n) , making the induction hypothesis identical with the re quirement f (rec(n + 1)) ? 0. Next we consider (5.3). Our proof demands that n = 1 be treated sep arately. We have rec(1) T(rec(O) , 0) T(c, 0), and this is simply b since f(c  d) f( b ) < 0. Therefore, the property (5.3) is identical with the as sumption (ii) . Now the proof for any hypernatural N ? 2. We consider the formula =
=
=
=
=
N(n) 1\ n :S: N  2 1\ rec (N  n) =/= rec(N  n  1)  d/2 N n  l
(5.4)
and consider two possibilities. First possibility: there are no hypernaturals n satisfying (5.4) . This means that rec(N  n)  d/2N n  l
=
rec(N  n  1)  d/2N n  2
for 0 :S: n :S: N  2 , and by example 5.2.3 it follows that rec(N)  d/2 N l = rec(1)  d = c.
( 5.5 )
As f(c) < 0, we conclude that (5.3 ) holds for our N. Second possibility: there are hypernaturals n satisfying (5.4) . If so, let no be the smallest one, as provided by theorem 5.2.6. Then no :S: N  2 and rec(N  no) =/= rec(N  no  1)  d/2 N nol l.e. T(rec(N  no  1), N  no  1) =/= rec(N  no  1)  d/2 N no l . The definition of T(t, n) shows that then, inevitably, T(rec(N  no  1), N  no  1)
=
rec(N  no  1),
meaning that f(rec(N  no  1)  d/2 N  no l )
=
f(rec(N  no)  d/2 N no l ) < 0. ( 5.6 )
By the leastness of no, rec(N  n)  d/2N n l = rec(N  n  1)  d/2 N  n  2
References
75
for 0 1' satisfying f(x) < 0, which by (5. 1) leads to f( r ') < 0 and contradicts the property (iii) for 1' · Likewise for the possibility 1 « 1' · Therefore 1' � 1. D This theorem allows us to equip ERNA with a square root up to infinitesi mals function. Example 5.2.4 For every finite constant p > 0, ERNA provides a constant
1 > 0, unique up to infinitesirnals, such that 1 2 � p.
=
It follows from the properties of an ordered field that the term f ( x) 2 x  p and the constants b = 0, c = 1 + p satisfy the requirements of theo rem 5.2.9 , including the extra requirement (5. 1). If 1 is the constant resulting from the theorem, then 1 2 � p and for every natural n � 1 there are x > 1 1/ n with x 2 < p. Moreover, x < 1 + p by the properties of the ordered field. Hence 1 2 < x 2 + 2x/n + 1/n 2 < p + 2(1 +p)/n+ 1/n 2 . By corollary 5.2.1 , we conclude that 0 ws t.he contributions of .Jose Sousa Pinto to t;his area up to his untimely death four years ago. Following the original presentation of nonstandard models for the Seba.stiao e Silva axiomatic treatment of distrilmt.ions and ultradistribut;ions he worked on a nonstan dard theory of Sato hyper:functions, using a simplC' ultrapower model of the hyp cr eals. ( Tllis in particular allows noustandru:d representations for gen eralised distributions , such as those of Rounlieu , B em·ling , and so 011.) He also considered a nonstandard theory for (;he generalised func tions of Colombeau, and finally tmned his attention to the hyperfinite repres entation of generali sed functions, following the work of Kinoshita. r
6.1
Introduction
Jose Sousa Pinto of the University of Aveiro, Portugal, died in A ugust
2000
after a prolonged and debilitating illness. His interest in nonstandard methods, particularly in their application to the s tudy of generalised functions, was of long st and ing and he will be especially remembered for his part. in t he organi
International Colloquium of Nonstandard Mathematics h eld at Aveiro 121 iu 1994. A most modest. aud unassuming mathematician, his contributions to NSA are less \vell known than their value
sation of the highly successful
deserves and this paper is concerned to report his work and to stand as some tribute to his memory. From a personal point of view
I would also wish
to take
· Department. of ElecLTonic and Electrical Engineering, Loughborough Uni versity, Lough boTough, U K .
[email protected] . com
6.2.
77
Distributions, ultradistributions and hyperfunctions
this opportunity to acknowledge the value and pleasure I have had in working
with him over many years.
6.1.1
Generalised functions and N.S.A.
The theory o f generalised functions is a subject o f major importance in
modern analysis a.nd one that inal presentation
! lSI
has gone
through many changes since the orig
of the theory of distributions in tho forrn given to it. uy
Laurent Schwartz in the
1950s.
Various alternative approaches to the theory
have been explored over the years, and the subject, has been expanded (and complicated) by the introduction of generalised distributions of several types,
ultradistributions, hyperfunctions and so on. In recent years the development of a unifying and simplifying treatment of the whole subject area has become
possible through the use of Nonstandard Analysis (N.S.A.) The applic a ti on of nonstandard methods to distributions and other generalised ftmctions was aheady considered by Abraham
1966, then. a
Robinson
in his classic text
[15!
on N.S.A. in
and various workers have extended and developed this approach since It
was
Sousa Pinto who first considered the possibility of developing
nonstandard realisation of the S e bas t i ao e Silva axioms for Schwartz dis
tributions
!6] ,
and later for nltradistributions
[7] .
His further work on Sato
hyperfunctions remained unpublished at the time of his death and an outline
of this forms the
main part of the present paper.
The first section of the paper briefly reviews standard material on distribu tions, ultradistributions and Sato hyp, V¢ E TJ.
of
6.
78
The Sousa Pinto approach to nonstandard generalised functions
It follows from this definition that all distributions are infinitely differentiable in this sense. Moreover, it can be shown that every distribution is locally a finiteorder derivative of a continuous function. Those distributions which are globally representable as finiteorder derivatives of continuous functions are called, not unnaturally, finiteorder distributions. The space of all such finiteorder distributions is denoted by Dfin (I�) . Every locally integrable function f defines a socalled regular distribution p,f according to < fLJ · ¢
>=
1+= f(x)cjJ(x ) dx, =
for all ¢ E D.
D' contains elements other than such regular distributions, so that D' is a proper extension of the space Ll oc (�) of all locally integrable functions. In this sense distributions may be legitimately described as generalised functions. However there is no direct sense in which a distribution can be said to have a value at a point. This becomes particularly clear in the case of those distribu tions which are not regular. The delta function is the prime example of such a singular distribution, being defined simply as that functional 6 (obviously linear and continuous) which maps each function ¢ E D into the number ¢(0) . It is a finiteorder distribution since it is the second derivative of the continuous function ( t) = tH ( t), where H denotes the Heaviside unit step.
6.2.2
x+
The Silva axioms
The definition of distributions as equivalence classes of nonstandard inter nal functions in a nonstandard universe was already made explicit in Abraham Robinson's original text [15] on N.S.A. Several other nonstandard models for D' (�) have since appeared. In particular such a model was presented by Hoskins and Pinto [6] in 1991, based on the axiomatic treatment of distribu tions given by Sebastiao e Silva [19] in 1 956. The Silva axioms for finiteorder distributions on an interval I C � can be stated as follows: Silva axioms for finite order distributions Distributions are elements of a linear space £ (!) for which two linear maps are defined: L : C (I) + £ (!) and D : £(!) + £(!), such that
f E C is a distribution) . corresponds Dv E £ such that, if v = ( j )
Sl L is the injective identity, (every S2 To each v E £ there then Dv = L ( j ' ) S3 For
L
v E £ there exists f E C and r E No such that v = Dr ( f ) . L
E C 1 (I)
6.2.
79
Distributions, ultradistributions and hyperfunctions
S4 Given j, g E C and r E No, the equality Dr L(j) only if ( ! g) is a polynomial of degree < r. 
=
Dr L(g)
holds if and
Silva gives an abstract model for this set of axioms as follows : define an equivalence relation o on No x C by ( r, f ) o ( s , g)
++ 3m E No { m ? r, 1\ (I;:'  r f  r;: s g ) E II m } s
where II m is the set of all complexvalued polynomials of degree less than m and I� is the kth iterated indefinite integral operator with origin at a E I . Now write Then C= is a model for the Silva axioms S lS4, and every model for the Silva axioms is isomorphic to C= . In particular Dfin (l�) is isomorphic to C= (�) . The extension to global distributions of arbitrary order is straightforward. See, for example, the exposition of the Silva approach to distribution theory given in Campos Ferreira, [3] . A nonstandard model for Coo A nonstandard model for these axioms was presented by Hoskins and Pinto [6] , using a simple ultrapower model *� = �N / for the hyperreals. It may be summarised as follows: The internal set *C= (�) is the nonstandard extension of the standard set coc (�) of all infinitely differentiable functions on �' rv
*Coc (�) = {F = [ ( !77 ) n EN] : fn E c = (�)
for nearly all
n E N}.
This set is a differential algebra. We denote by 5C(�) the (external) set of all functions F E *C= (�) which are finitevalued and Scontinuous at each point of *�b· An internal function F E *C = (�) is then said to be a predis tribution if it is a finiteorder *derivative of a function in 5C(�). The set of all such predistributions is given by,
r 2:0 r = * * (�) : F for some E 5C(�) and some r E N0 }. = {F E c = D We then have the following (strict) inclusions: The members of *D={5C(�) } arc the nonstandard representatives of finite or der distributions on �. Given two such predistributions F and G, we say that
80
6. The Sousa Pinto approach to nonstandard generalised functions
they are distributionally equivalent, and write F3 G, if and only if there exists an integer m E No and a polynomial Pm of degree m (with coefficients in *JR) such that
*T;:(F  G) � Pm
where *I;;' denotes the mthorder *indefinite integral operator from a E *Rb· Then for any F E *Doc{5C(IR) } we denote by /LF = 3 [ F] the equivalence class containing F and call it a finite order Bdistribution. The set of all such equivalence classes is denoted by
3CDO (IR) is a nonstandard model for the Silva axioms and is isomorphic with Djm(IR). 6.2.3
Fourier transforms and ultradistributions
For the classical Fourier transform of sufficiently wellbehaved functions we have
j+= j+= ixy = f(y)e dy f(y) = 217r ](x)eixy dx and the Parseval relation j+= j+=  = f(x)g (x)dx = DO ](y)g(y)dy. ](x) =
 CXl
To extend the definition of Fourier transform to distributions Schwartz used a generalised form of Parseval relation to define jl as the functional satisfying
jl , cjJ >=< P,, cjJ > . The difficulty here is that if cjJ E D then its Fourier transform J; belongs not to D but to another space Z Z (IR) which comprises all those functions 'ljJ such that '1/J(z) is defined on CC as an entire function satisfying an inequality
O where 5D is the *CCb submodule of 5C comprising all infinitely *differentiable
functions of hypercompact support which are finitevalued and Scontinuous on *�b· If F = [(Jn ) n EN] is any internal function in *D then its inverse Fourier transform F = *F1{F} is defined in the obvious way as F = [( fn ) n EN] = [(F1{fn} ) n EN] and it follows readily that F E *D if and only if F = .r1{ F} E *Z. Not every internal function in *D represents a 3distribution and similarly not every internal function in *Z represents an ultradistribution. However the following result was established in [7] . Let H(CC) denote the space of all standard complexvalued functions which can be extended into the complex plane as entire functions. For each entire function A(z) = L �=O an zn in H(CC) define the ooorder operator A : Z + Z by setting, for each ¢ E Z,
CXl
CXl
n=O
n= O
Then:
The inverse Fourier transform of a 3 distribution in 3C= (�) is representable as a finite sum of (standard) xorder derivatives of internal func tions in *Z whose standard parts are continuous functions of polynomial growth.
Theorem 1
6.2.4
Sato hyperfunctions
Another approach to the required generalisation of the Fourier transform stems from the work of Carlemann [4] . He observed that if a function f, not necessarily in £1 (�), satisfies a condition of the form
fox l f(y) l dy = O( l x i K )
for some natural number "" '
and if we write
and
82
6. The Sousa Pinto approach to nonstandard generalised functions
then g 1 ( z ) is analytic for all "S(z) > 0 and g 2 ( z ) is analytic for all "S(z) < 0. Moreover, for (3 > 0, the function
g (x) ::::: gl (x + i(J)  g 2 (x  i(J) is the classical Fourier transform of the function e f31 t l f ( t). The original func tion f can be recovered by taking the inverse Fourier transform of g and multiplying by ef31 t 1 . This suggested that a route to a generalisation of the
Fourier transform could be found by associating with f a pair of functions h (z) and h (z) analytic in the upper and lower halfplanes respectively. This idea forms the basis of the theory of hyperfunctions developed by M. Sato [17] in 1959/60, (although it was anticipated by several other mathematicians) . In order to give a brief sketch of this theory it is convenient to introduce the following notation: H(C\�) = the space of all functions analytic outside the real axis. H(C) subspace of all functions in H(C\�) which are entire. Hp , l oc (C\�) space of all functions e in H(C\�) which are of arbitrary growth to infinity but locally of polynomial growth to the real axis (that is, such that for each compact K C � there exists CK > 0 and rx E No such that =
=
for all z E C with �(z) E K and sufficiently small "S(z) i= 0) .
The hyperfunctions of Sato are the members of the quotient space Hs (�) H(C\�) /H(C), that is, the set of all equivalence classes [B] , where B(z) is defined and analytic on C\� and e1 B2 iff B 1  B2 is entire.
Definition 2
=
rv
Sato hyperfunctions constitute a genuine extension of Schwartz distributions. This is shown by the following crucial result established by Brcmermann [1] , in 1965:
If p, is any distribution in D' then there e.rists a function p,0 (z) defined and analytic in C\� such that
Theorem 3 (Bremmerman)
Conversely, if B E Hp , l oc (C\�) then there exists p, E D' such that B(z) p,0 (z) . Identifying p, E D' with [B] E Hs (�) gives an embedding of D' such that the mapping S D' Hp ,l oc (C\�) is a topological isomorphism. =
:
Remark 4
itly by
)
Note that if p, has compact support then p,0 (z) is given explic
6.2. Distributions, ultradistributions and hyperfunctions
p,o (z)
=
1 2 .

Jr Z
< p,, ( x  z)
83
1>.
For example, 6° (z) = 2;; < 6, ( x  z) >=  2;iz and we have < 6, ¢
6.2.5
E
> = lim 
c:;.0
7r
1+=  CXl
X2
1 ¢ ( x ) dx = ¢(0) . + E2
Harmonic representation o f hyperfunctions
For each Sato hypcrfunction [e] in Hs (�) we can choose some specific function e 0 E [e] as the defining function of the hyperfunction and write X
Then e1[ maps the halfplane rr+ = � �+ into c, and is harmonic on rr+ . Define H (II + ) to be the linear space of all (real or complexvalued) functions defined and harmonic on rr+ , and let r H(C\�) + H (II+) denote the map given by :
Then we have the result
r is an onto map and if r(e 0 ) = r(v0 ) then 0 0 e  v is a complex constant. Now suppose that [e] is the null hyperfunction, so that e0 belongs to H(C) . It is easily shown that e1[ ( x, y) is an entire function in both variables (that is,
Theorem 5 (Li BangHe, [13] )
can be extended into C x C as an entire function) , is odd in the variable y and such that On the other hand we have immediately from the above theorem,
Let e1[ E H(II+) be an harmonic function entire in both variables and odd in the variable y. Then there exists an entire function e E H(C) such that e1[ ( x , y) = e ( x + i y)  e ( x  iy) for all ( x. y) E rr+ . Corollary 6
Denote by H o ( II+ ) the linear subspace of H(II+) comprising all functions which extend into C x C as entire functions in both variables and which arc odd in the second variable. Then we have Hs (�)
rv
H (II + ) / H o (II+ )
84
6. The Sousa Pinto approach to nonstandard generalised functions
and every equivalence class [B1r (x, y)] E H (II + )/Ho (II + ) is a representation by harmonic functions of the corresponding hyperfunction [B] E Hs (�) . In the sequel we will use analytic or harmonic representation for hyperfunctions as occasion demands.
6.3
Prehyperfunctions and predistributions
Nonstandard representation of hyperfunctions In the present context w E * � denotes the infinite
hypernatural number defined by [(n) n EN] in *Noc . Then to each harmonic function V1r E H(II + ) there corresponds an internal function F{ v } : * � + * C defined by where c; = w  1 E mon(O) and *v1f *(v1f) is the nonstandard extension of v1r · The internal function F{ v } clearly belongs to *C = (�) . If we define a map w : H(II + ) *C = (�) by setting w(v1f) = F{ v } then we have =
____,
where the inclusion is strict. The set w 'Hs = w (H (II + )) is an external subset of *C = (�) ; however it can be embedded into an internal subset as follows: Consider the nonstandard extension *H(II + ) of H(II + ) comprising all *harmonic functions on *II + and then define,
Then we have =
where w H( �) contains elements which are infinitely close to members of w 'Hs w (H (II + )) and also elements which are far from any internal function in that space. The members of w H ( �) will generally be called prehyperfunctions.
Prehyperfunctions which are near some element of w 'Hs (�) are said to be nearstandard prehyperfunctions, and the others may be called remote prehyperfunctions . The set of all such nearstandard prehyperfunctions will be denoted by The clements of w H( � ) enjoy an important property which is not shared by all internal functions in *C = (�) , namely:
6.3. Prehyperfunctions and predistributions
85
Every prehyperfunction in w H(l�) on the line may be extended into the hypercomplex plane as a * analytic function in the infinitesimal strip D10 = {z E * CC : l lm(z) l < c } .
Theorem 7
Any function is analytic at the centre of an open disc on which it is harmonic. Hence any internal function F(x) , x E *R in w H(l�) extends into the hypercomplex plane z = � + iT) and is *analytic on the disc Proof.
2
T) 2 < 62 . For � = x we have < T) < c and since x may be any hypcrrcal it follows that the internal function F{ v} ( x) extends as a *analytic function into the (�
_
X)
+
c:
infinitesimal strip in *CC defined by I Im(z ) l < c. D The converse docs not hold: not every *analytic internal function in the infinitesimal strip 010 is a prehyperfunction on the line. The product of two prehyperfunctions, for example, is a *analytic function on the strip nc but need not itself be a prehyperfunction: w H (l�) is a linear space over *CC but not an algebra. Now let A(D10) denote the set of all internal functions in *CDO (�) which may be extended *analytically into the strip nc. A(Dc) is a differential subalgebra of *CDO (�) with respect to the usual operations of addition, multiplication and *differentiation. Moreover we have, The product of any two prehyperfunctions makes sense within the algebra
A(D10) although such a product will not generally be a prehyperfunction. In view of the above inclusions it seems appropriate to call the members of A(D10) generalised prehyperfunctions.
Finally let 0(010) be the subset of all internal functions 8 E A(D10) such that *D k 8(x) � 0 for all x E *�b and for all k E No . 0(010) is a linear *CCbsubmodule (but not an ideal) of A( 010) and therefore is a module over *CCb: its elements might conveniently be called generalised Every such generalised 3 hyperfunction which may be represented by an internal function in w H ns (�) is called a standard 8hyperfunction or simply a 8 hyperfunction. The set of all 3hyperfunctions on the line is defined by 8hyperfunctions.
and we have the isomorphism
6. The Sousa Pinto approach to nonstandard generalised functions
86 6.4 6.4. 1
The differential algebra A(fl�:: ) Predistributions of finite order
Although not every continuous function f on � may be continued ana lytically into the complex plane, every such function does admit an analytic representation in the sense that there exists a unique hyperfunction [f ( x, y)] in Hs (�) such that f1f (x, y) + f (x) as y l 0 uniformly on compacts. Then the internal function F(x) *f1f (x, ) belongs to wHs (�) C w Hns ( �) C A(Slc:) and is such that for all x E * �b· F(x) � *f(x) , F(x) is Scontinuous at every point (standard and nonstandard) of *�b; that is, Vx, y E * �b [x � y =? F(x) � F(y)]. Reciprocally, every internal function F E A(010) which is finite and S continuous at every point x E *�b is infinitely close to a (standard) continuous function f defined on �. We denote by 5C (0c:) the subalgebra of all functions in A(010) which are finite and Scontinuous on *�b· An internal function F *� + CC is said to be finitely *differentiable at x E *� if it is *differentiable at x and, in addition, *DF(x) is a bounded number. 7r
=
c
:
8 An internal function F E A( 010) which is finitely * differentiable on *�b belongs to 5C (S1c:) .
Theorem
For any internal functwn F in 5C(S1c:) the standard function f = st (F) will be called the shadow of F while the regular distribution vp = stv (F) E D' (�) generated by F will be called the D shadow of F.
Definition 9
Now let be an arbitrary function in 5C(010) . Since 5C(0c:) is not a differential algebra the derivative, *D, will not necessarily belong to 5C (S110). However it is easy to see that the functional /L*Dif> D(�) + CC defined by :
< /L *Dif> , cjJ > = st( < ,  *¢' >) is a well defined distribution. Hence *D has a welldefined Dshadow. If, in addition, *D itself belongs to 5C (Slc:) then V*Dif> is the (regular) distribution generated by the standard function f' = st (*D) which is the standard deriva tive of the function f st () . Let *D0{5C (010) } denote the set of derivatives of functions in 5C (0c:) and for each k E No define =
6.4.
The differential algebra A(Dc:)
87
5 C (De:) . We can now define the following (external) 00 5 Coo (D c: ) U *D k { 5 C(D s ) } . k=O Then for each F E 5C00 (Dc:) there exists an Scontinuous function E 5C(Dc:) and an integer r E No such that F *Dr. The functional /LF D(IR) , defined by where *D0{8C (D10)} subset of *H (D10) :
=
=
:
is a distribution. We call /LF the Dshadow of the internal function F and write fLF = stv (F) . That is to say, we extend stv into 8Coo (Dc:) as a map ping with values in D' (IR) . An internal function F E *H(Dc:) is said to be *Svdifferentiable if, for every ¢ E D there exists a standard number b¢ such that < T  1 {F(x + T)  F (x) }, *¢ >� b¢ for all T � 0, T # 0. As is easily confirmed, every function F in 5C00 (De:) is *Svdifferentiable, and we define the 5 differential order of F to be the number 5o( F) defined by 5 o(F)
=
min{j E No : F
=
*DJ , for some E 5C (D10)}.
Accordingly we call 5C00 (Dc:) the set of all predistributions o f finite
8 differential
order.
Replacing the (standard) concept of distribution of finite order by the (nonstandard) concept of predistribution of finite 5 differential order it is clear that 5Coo (Dc:) with the *D operator constitutes a natural (nonstan dard) model for the axiomatic definition of distributions of finite order given by J .S. Silva. Distributional equivalence
We may now glue together all internal functions in 5C00 (De:) which have the same distributional shadow. That is to say, we define the following equivalence relation on 5 Coo (De:) : P G E 5C00 (D10 ) are distributionally equivalent , written F3G, if and only if they have the same distributional shadow. The quotient space is a *Cbmodule which is isomorphic to the space C00 (IR) of .J .S. Silva distribu tions of finite order.
88
6.4.2
6.
The Sousa Pinto approach to nonstandard generalised functions
Predistributions of local finite order
Finite order predistributions in 5C= (�) are not the only prehyperfunc tions which have a distributional shadow. Let F E w1is (�) \5Coc (D s ) be a prehyperfunction such that for every compact K C � there exists an integer rx E No and an internal function W K E w1is (�) which is Scontinuous on some *neighbourhood of *K so that
for all x E *K C �b · The smallest such rx will be called the 5differential order of F on *K, and denoted 5oK (F) . If ¢ E D (R) has support contained in K then < F, * 7r >=< K, (  l rK *c/J (r K ) > is a bounded number and so F has a distributional shadow in DK (�) C D (�) . Since K may be any compact in R it follows that F has a shadow in D (R) and so p, = stD (F) will be a well defined (standard) distribution in D' (R) . Denote by 5C1f (�) the subset of all prehypcrfunctions which have a distri butional shadow. Then we have the inclusion
}
{
Moreover 5C1f (�)\5Coc (�) is not empty since it contains, for example, the internal function F : *� *C defined by ____,
(
F x)
=
+=
� *D"
1 w 1r l + w2 (x  i) 2
The members of 5C1f (�) will be called predistributions of local finite order or, more simply, predistributions. 6.4.3
Predistributions of infinite order
Let be any internal function in 5C (S110) and suppose that there exists an harmonic function g E H (II + ) such that (x) = *g ( x , ) Let also r = [rn] be an arbitrary infinite hypernatural number. For every n E N the function c:
arn g aXrn (x, y),
.
( x, y) E rr +
is again harmonic on rr + , and so *Dr is a generalised prehyperfunction in *1i(Sl10) . The internal function *Dr is locally bounded by r! wr ; that is to say,
6.4.
The differential algebra A(Dc:)
89
for each compact K C � there exists a bounded constant CK such that for all x E *K we have In general the infinite order derivative *Dr may have neither an ordinary shadow nor a distributional shadow, and stv (*Dr ) may have no meaning. On the other hand, for any ¢ E D(�), we have
< *Dr , *cjJ > = ( 1Y < , *cjJr > . But, since there exist test functions in D(�) whose derivatives may grow arbi
trarily with the order, then
will not in general be a bounded hypercomplex number. Hence stv (*Dr ) may have no meaning. However we can define a family of standard part maps which allow us to attach a type of shadow to derivatives of the form *Dr for infinite r E *N= and internal functions in 5C(D10). These standard part maps are defined on certain subspaces of D(�) , for example on those of socalled Roumieu type which we recall very briefly as follows. Spaces of Roumieu type [16]
Let M denote the set of all positive real sequences (Mp)pENo such that (a) (Mv) 2 ::; Mv l Mp+ l , p = 0, L . . . ,
(b) Mp ::; AhP min o ::;q::;p{MpMp q } , p = 0, 1 , . . . , for some positive constants A and h, += (c) L (Mp)  l /p < + oo. Further, let 1J(Mp) (�) be the subset of D(�) comp=O prising all functions ¢ whose derivatives satisfy
p = 0, 1, . . .
'
for some sequence ( lvfp) E M, where A and h are positive constants (generally dependent on ¢) . It can be shown that 1J(Mp) (�) is not empty for every sequence {Mp }v ENo E M . In particular, if there exists p E No such that Mp = + oo for all p � Po then {Mp}vENo belongs to M and 1J(Mp) (�) D(�). The space 1J(Mp) (�) is the union of the family of spaces =
6. The Sousa Pinto approach to nonstandard generalised functions
90
where K runs over the set of all compact subsets of � and h runs over all positive numbers. For each real number h > 0 and compact K C � ' vV(r, ) (�) , contains all functions ¢ E D ( Mp ) (�) with support contained in K and satisfying the above inequality for that particular value of h > 0. Each space vV(r, ' l is a Banach space with respect to the norm ll c/J I I v(Mp ) K,h
= sup p 2' 0
{ h M1 :rEsupK ¢ l (x) } , P
p
I (P
l
and D ( Mp ) ( �) is provided with the inductive limit topology. D' ( Mp ) denotes the topological dual of D ( Mp ) (�) and its elements are some times called generalised distributions in the sense of Roumieu. A linear functional is in D ( Mp ) (�) if and only if it is continuous on that space for every h > 0 and compact K C �.
6.5
Conclusion
The above outline of Sousa Pinto ' s nonstandard treatment of hyperfunc tions is reported in greater detail in [10] . Sousa Pinto ' s later work, developing the hyperfinite approach to distributions initiated by Kinoshita, is given in [8] and [9] , but most comprehensively in his last publication, the book [20] , which is now available in an English translation.
References
[1] [2] [3] [4] [5]
Distributions, Complex Variables and Fourier Tmnsforms, Massachusetts, 1965. N . J . CUTLAND ET AL ., Developments Nonstandard Mathematics, Longman, Essex, 1995. J. CAMPOS FERREIRA, Introdur;ao a Teoria das Distribuir;oes, Calouste Gulbenkian, Lisboa 1993. T. C ARLEJ\IANN, L 'integmle de Fourier et questions qui s 'y mtachent, Uppsala, 1944. B . FISHER, "The product of the distributions x r and c) ( r l ) ( x ) ", Proc. Camb. Phil. Soc., 72 ( 1972) 201 204. H . J . B REMMERMANN,
m
[6] R . F .
HoSKINS and J. SousA PINTO, "A nonstandard realisation of the .J .S. Silva axiomatic theory of distributions", Portugaliae Mathematica, 48
(1991) 195216.
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[7] R.F. HoSKINS and
J . SousA PINTO, "A nonstandard definition of finite order ultradistributions", Proc. Indian Acad. Sci. ( Math. Sci. ) , 109 (1999)
389395.
[8]
R .F.
HoSKINS and J. SousA PINTO, "Hyperfinite representation of dis tributions", Proc. Indian Acad. Sci. ( Math. Sci. ) , 1 1 0 (2000) 363377.
[9] R . F . HoSKINS and .J . SousA PINTO, "Sampling and ITsampling expan sions", Proc. Indian Acad. Sci. ( Math. Sci. ) , 110 (2000) 379392. [10] R . F . HoSKINS and J . SousA PINTO. Theories of Generalised Functions, Horwood Publishing Ltd., 2005. [11] KINOSHITA MoToo, "Nonstandard representations of distributions, I", Osaka J. l\Iath., 25 ( 1988) , 805824. [12] KINOSHITA l\IoT00, "Nonstandard representations of distributions, II", Osaka J. l\Iath., 27 ( 1990) 843861 . [13] L I BANGHE, "Nonstandard Analysis and Multiplication of Distribu tions", Scientia Sinica, XXI5 ( 1978) 561585. [14] LI BANGHE, "On the harmonic and analytic representation of distribu tions", Scientia Sinica ( Series A ) , XXVIII9 (1985) 923937. [15] A. RoBINSON, Nonstandard Analysis, North Holland, 1966. [16] C . RouMIEU, "Sur quelques extensions de la notion de distribution", Ann. Scient. Ec. Norm. Sup., 3e scrie (1960) 41121 [17] M. SATO, "Theory of Hyperfunctions", Part I , J. Fac. Sci. Univ. Tokyo, Sect. I, 8 (1959) 139193 ; Part II, Sect. I, 8 (1960) 387437. [18] L. ScHWARTZ, Theorie des Distributions I, II, Hermann, 1957/59. [19] J . S . SILVA, Sur l ' axiomatique des distributions et ses possible modeles, Obras de Jose Sebastiao e Szlva, vol III, I.N.I.C., Portugal. [20]
J. SocsA PINTO, Metodos lnfinitesimais de Analise Matematica, Lis boa, 2000. English edition: Infiniteszmal Methods of Mathematical Analysis, Horwood Publishing Ltd., 2004
N eut rices in more dimensions Imme van den Berg*
Abstract
the nonst.andm·d real number system, external sets. They may also he viewed a.s modules over the external set of all limited numbers, as such nonnoetherian. B
7.1.3
· · ·
Structure of this article
In section 7.2 we define formally the notion of neutrix and state the de composition theorem for neutrices in 2dimensional space. The decomposition theorem is proved in section 7.3. The actual proof, which is contained in section 7.3.3, needs some elementary external geometry (section 7.3.1) and algebra (section 7.3.2). In particular we study two kinds of division for neutrices in one dimension, and their relation to certain geometric properties of neutrices in two dimensions. We give special attention to practical aspects, like the calculation of the divisions, and present many examples. In general, the proof is neither fully algebraic, nor fully analytic. Instead, it tends to be a mixture of algebraic and analytic arguments, where typically algebraic operations arc adapted to the order of magnitude of the quantities involved. 7.2
The decomposition theorem
Definition 7.2.1
subgroup of �k .
Let k E N, k � 1 be standard. A neutrix is a convex additive
A convex (external) subset N of � is a subgroup if and only if it is sym metric with respect to 0 and whenever x E N one has 2 x E N. Then by external induction nx E N for all standard n E N; by convexity, one has lx E
96
7.
Neutrices in more dimensions
N for all limited l E R This indicates an alternative way to define neutrices: they should be modules over £, the external ring of all limited real numbers: ·
a subset N of � is a neutrix if and only if £ N = N . The only internal neutrices of �k are its linear subspaces. Thus the only internal neutrices of � are {0} and � itself. Two obvious external neutrices in � arc £ itself and 0, the external set of all infinitcsimals. The ncutriccs of the form c£ for some positive c E � ("cgalaxies") are isomorphic to £ and the ncutriccs of the form c0 for some positive c E � ("chalos") arc isomorphic to 0 . Every neutrix N f. {0} is nonnoetherian in an external sense: there exists always a strictly ascending chain of subneutrices (Nn ) n EN with N1 � N2 � for standard indices n. Indeed, let w E � � Nn � Nn + 1 � be positive unlimited, and put Nn = wn £. Then Nn ct Nn+ 1 for all indices n. Consider ww which is not an element of Nn for all standard indices n. Let c E N be sufficiently small such that also cww E N . Then (cNn ) st n is a strictly ascending chain of £submodules of N . It may be proved [5] that for any strictly ascending chain ( Nn ) st n of neutrices, the union U st n Nn is neither isomorphic (for internal homomorphisms) to £ nor to 0. This suggests that there is a rich variety of external neutrices in �' nonisomorphic with respect to internal homomorphisms. Still, as it is tacitly understood that a neutrix is defined by a bounded formula ¢ of Internal Set Theory, a neutrix has a simple logical form, for ¢ may be supposed to be internal, galactic or halic [5] . The main theorem of this paper asserts that in a sense augmenting the dimension to two docs not generate entirely new types of ncutriccs, for any neutrix in �2 may be decomposed into two neutrices of R We adopt the notation Nx { nx I n E N} for the ncutrix of all multiples of some vector x with coefficients in some neutrix N C �. Theorem 7.2.2 (Decomposition theorem) Let N C �2 be a neutrix. Then there are ne1drices N1 ::) N2 in � and orthonormal vectors u 1 , u 2 such that · · ·
· · ·
=
Moreover, if lvf1 ::) lvh in � are nev.trices and 111 , v 2 are O'rthonormal vectors with N M1 v1 EB M2 v2 , it holds that lvh N1 , M2 N2 . =
=
7.3
=
Geometry of neutrices in IR 2 and proof of the decomposition theorem
The present section is divided into various subsections, conform the various stages of the proof of the decomposition theorem.
7.3. Proof of the decomposition theorem
97
The first subsection contains the definitions of the notions of thickness, width (smallest thickness) and length (largest thickness) of neutrices, which are the basic ingredients of the proof. We prove an important theorem, called the "sectortheorem", expressing convexity of the thicknesses of neutrices in two dimensions on not too large sectors. One of the consequences is that the thicknesses in most directions arc minimal. This implies that. the width of a neutrix is realized in some direction, so in the decomposition represented in the main theorem we obtained already the ncutrix N2 . Other important notions used in the proof are nearorthogonality and nearparallelness. The most important step in the proof of the decomposition theorem in JR2 consists in establishing a direction, which realizes the length of the neutrix; then up to a rotation, the neutrix will simply be "length times width". This part of the proof uses a form of euclidean geometry in which the points and lines may have nonzero thickness, in fact such a thickness takes the form of a neutrix. We have to adapt some definitions, operations and theorems of ordinary, exact euclidean geometry to this geometry of "clouds" or "mistbanks". This is done in the first subsection, while the second subsection contains an algebraic tool: the "division" of a neutrix by another, the result of which may be calculated through a "subtraction", after taking logarithms. The final subsection establishes the existence of a direction which realizes the length. It uses an argument of "external analysis": every (external) lower halftine has a supremum, which is an external number, i.e. , the sum of an ordinary real number and a ncutrix (theorem 7.3.23) . In a sense, the external set of directions which realize the length is the supremum of an external set of directions which do not realize the length. 7. 3. 1
Thickness, width and length of neutrices
Let N c IR2 be a neutrix. We call N square in case there exists a neutr"ix lvl C lR such that N = M x AI . Definition 7.3.2 Let N C IR2 be a neutrix and r E IR2 be a nonzero vector. The thickness of N in the direction of r is the neutrix Tr of lR defined by Tr E JR EN . Definition 7.3. 1
=
{X I X nfu ·
The width W of N is defined by
w
and its length L by
=
n
rES1
Tr ,
}
98
7.
Neutrices in more dimensions
In an obvious way, if X is a subset of �2 one may define II X II = So, if r is unitary, we may write the thickness in the di rection of r alternatively as Tr = ± li N n �r l l · As an example, consider the neutrix £ X 0 C � 2. Let r ( ���: ) . Then Tr £ if cjJ 0 (mod ) otherwise Tr 0 . Hence W 0 and L £. It will be shown later on that for every ncutrix the width is assumed in some direction (in fact most of the directions) and the same holds for the length (in only few directions) . For square neutrices N lvf2 all thicknesses arc equal to M, hence their width and length arc also equal to M. { llxll l x E X }. =
=
=
=
=
"':'
1r
,
=
Definition 7 . 3 . 3
Let N
C �2 =
be a ne'Utrix and x, y =
E �2 .
We call x and y
nearly orthonormal if l l x l l II Y II 1 and < x, y >"'=' 0. We call x and y nearly orthogonal if l � l and l l � l l are nearly orthonormal. Let 0. Then the vectors ( 6 ) and ( v'l':_c:2 ) are nearly orthonormal. The 0 1 1 vectors ( 6 ) and ( i ) are nearly orthogonal. We introduce now a notion of nearorthogonality with respect to neutrices. c "'='
Definition 7.3.4 Let N C �2 be a nonsq'Uare ne'Utrix and x, y E �2 be non zero vectors. We call a line �y nearly parallel to N if Ty > W. We call x a nearnormal vector of N if X is neaTly orthogonal to all y E S 1 S'Uch that y is nearly parallel to N.
As an example, consider the neutrix N £ x 0 C �2 . The vector ( � ) is nearly orthogonal to N . Indeed unitary vectors Yc: such that y is nearly parallel, i.e. Ty" £ > 0 , are all of the form ( v'l':_c:2 ) with "'=' 0. Then ( ( n . ( v'l; c:2 ) ) 0. Note that all vectors of the form ( 1 .+ a ) with a , (3 0 are also nearly orthogonal to N . The notions of nearparallelness and nearnormality will be justified for neutrices N in two dimensions by theorem 7.3.6. In fact it is a consequence of the next simple, but important theorem, based on the convexity of N, that most directions have the same thickness. This proves a fortiori the existence of a direction which realizes the width. �
c
=
"'='
"'='
(Sectortheorem) Let N C �2 be a ne'Utrix and a and b two 'Unitary vectors which make an angle e with 0 ::; e :ii Let c be a 'Unitary vector, which makes an angle I with a S'Uch that 0 ::; I ::; e. Then
Theorem 7.3.5
1r.
Proof. Without restriction of generality, we may assume that Tb ? Ta > 0 and that a = ( 6 ) . Let x E �a n N . Consider the change of scale M = Nj l l x ll ·
7.3. Proof of the decomposition theorem
99
The image of x is a E M. Notice that also b E M, for Tb ? Ta . Let ( be the intersection of the lines ab and �c. Then ( E M by convexity and 1 1 ( 11 is appreciable. Because £( E M it holds that (/ 11(11 E lv1. Put z = ll x ll · ( / 11(1 1 . D Then z E N. Because ll z ll = ll x ll we conclude that Tc ? Ta . Theorem 7.3.6 Let N C �2 be a neu.trix. Then theTe e:r:ists an unitary vector u such that Tu = W. In fact the set of directions corresponding to an unitary vector r with T,. = W contains an interval of the fonn ( a + 0 , a + 1r + 0 ) .
(
)
Let M = min Tm , Tm . By the sectortheorem Tu ::;:,. M for all unitary u in the first quadrant. But Tm = T( ;} ) and Tm = T( �1 ) hence Tu ? M for all unitary u in every quadrant. Hence !vi = W. If N is square, one has T,. = W for all unitary vectors r. If not, we may rotate N in order to obtain that TW = T(01 ) > W. Let r be an unitary vector which makes an angle e with the horizontal axis such that 0 � e � If T,. > W, one should have Tm > W, by the sectortheorem applied to [B, 1r] or [0, B] , depending to whether e � � or e ? � . This implies a contradiction, hence T,. = W. Thus we proved the second assertion,with a = 0. D Proof.
Jr .
Let N C �2 be a neutrix with width W. Let u, v be two orthonormal vectors. Then Tu = W or Tv = W. The proof of the decomposition theorem for a neutrix of �2 is easy, once
Corollary 7.3. 7
we know that its length is realized in some direction.
Let N C �2 be a neutrix with length L and width W . Assume there exists a unitary vector u such that N n �u = Lu. Let v be unitary such that u _l v . Then N = L u EB W v . Moreover, if u', v' are orthonormal and N1 , N2 C � are neutrices with N1 ::) N2 such that N = N1 u' EB N2 v', one has N1 = L and N2 = W . Proof. By corollary 7.3.7 it holds that Tv = W. Because N is a neutrix, we have Lu EB Wv C N. Conversely, let n E N. Let p be the orthogonal projection of n on �u. Because II n il E L and II P II � II n i l one has II P II E L, so p E N. Then n  p E N, and because Tv = W, it follows that n  p E Wv. Hence n = p + (n  p) E L u EB W v and N C L u EB W v . We conclude that N = L u EB Wv. We prove now the uniqueness part. The neutrix N cannot contain vectors larger than its length, so N1 C L. If there exists A E L \ N1 , all vectors n in N satisfy ll n ll < 1 > 1 , so L cannot be the length of N. So L ::) N1 , from which we conclude that L = N1 . By corollary 7.3.7, one has N2 = W. D Theorem 7 .3.8
100
Neutrices in more dimensions
7.
It is now almost straightforward to prove the decomposition theorem for neutrices in �2 if their length is of the form A£. =
Let A E � ' A > 0. Let N be a neutrix with length L A£ and width W. Then there are orthonormal vectors u and v such that N = Lu EB Wv .
Proposition 7.3.9
=
By rescaling if necessary we may assume that A 1 . Let u E N be unitary. Then £ u E N because N is a ncutrix, and N n �u C £ u because the length of N is £ . Hence N n �u = £ u = Lu . Let v be unitary such that u _l v. Then N Lu EB W v by theorem 7.3.8.0 The decomposition theorem is also easily proved for subneutriccs of a given neutrix with length less than the length of this neutrix. Indeed, we have the following definition and proposition.
Proof.
=
Definition 7. 3 . 1 0 Let N C �2 be a neutrix with M C � be a neutrix with W C M C L . We define
length L and width W. Let
NM = {n E N l ll n ll E M } . Clearly NM is a neutrix with length M. Its length is realized in any direc tion for which N contains a vector u with Tu ;:::: M. Then the next proposition is a direct consequence of theorem 7.3.8. Proposition 7.3. 1 1 Let N C �2 be a neutrix with length L and width W . Let A1 C � be a neutrix with W C lvl C L . Assume there is a unitary vector u such that Tu ;:::: M . Let v be a u.nitary vector v such that ( u, v) is orthonormal.
Then NM
=
Mu EB Wv .
Let N C �2 be a neutrix. We call N lengthy if it is not square, and if its length is not of the form L = s£ for some E �� > 0.
Definition 7. 3 . 1 2
c
c
So the twodimensional decomposition theorem will follow once we have proved that lengthy neutrices assume there length. Note that the proof of proposition 7.3.9 does not work, because for every A E L there exist n E N such that II n il = cj:J · A. In order to prove that a lengthy neutrix assumes its length too, we work "from outside in". We consider lines with unitary directions u such that Tu < L, with u in an appropriate segment of the unit circle, in such a way that all lines "leave to the right". We divide the segment into two (external) classes: directory vectors for lines which are "leaving upside" and directory vectors for lines which arc "leaving downside". It follows from the completion argument mentioned earlier that the two classes do not entirely fill
7.3. Proof of the decomposition theorem
101
up the segment, just as two disjoint open intervals within some closed interval omit at least one point. The unitary vectors left out are then exactly those directions v such that Tv = L. We introduce an appropriate scaling and orientation for lengthy neutrices, and make also precise what we understand by "leaving to the right upside" and "leaving to the right downside".
Let N C �2 be a lengthy neutrix. We call N appropriately scaled if W � 0 and L � £. Proposition 7.3. 14 A lengthy neutrix N C �2 is homothetic to an appropri ately scaled neutrix.
Definition 7.3. 13
Let A, w E L \ W be positive such that Ajw ::: 0. Consider M = Njw. Its width is Wj w C W/A C 0 and its length is Lj w which contains at least some unlimited elements. If Wj A � 0 we are done. If Wj A 0 we have D Wjw � W/ A, so clearly Wjw � 0. Hence M is appropriately scaled. Proof.
=
Let N C �2 be a lengthy appropriately scaled neutrix with length L . We call N appropriately oriented if
Definition 7.3. 1 5
Let N C �2 be a lengthy appropriately scaled neutrix. There exists a rotation p of the plane such that p(N) is appropriately oriented.
Lemma 7.3 . 16
Let W � 0 be the width of N. Then by proposition 7.3.11 there are orthonormal vectors u and v such that N£ = £u EB W v. So let p be a rotation D such that p(u) = ( 6 ) . Then p(N)£ = £ ( 6 ) EB W ( � ) .
Proof.
Let N C �2 be a lengthy appropriately scaled and oriented neutrix with length L and width W . Let A E L, A � 0. Then there exist orthonormal vectors u ::: ( 6 ) and v ::: ( � ) such that N£ .\ = £Au EB W v.
Lemma 7. 3 . 1 7
I f A is limited, the property follows from definition 7.3. 15. I f A is unlimited, any element n E N with ll n ll = A is of the form n = ( 6� ) with � ::: 00 and E ::: 0. Take u = vl�c2 ( � ) and v = vl�c2 ( 7 ) . Then the proposition follows from lemma 7.3.16. D In the final part of this section we consider points and lines close to a lengthy appropriately scaled and oriented neutrix. Proof.
102
7.
Neutrices in more dimensions
Let N C �2 be a lengthy appropriately scaled and oriented neutrix with and length L and width W . Let q E �2 be such that llqll E L. Then q is said to be infinitely close to N if q ::::: n for some element n E N. Let u ::::: ( 6 ) , v ::::: ( ? ) be orthonormal vectors such that N£>.. = £Au EB W v for some unlimited A with A � ll qll . Then q E,u + T)V with T) ::::: 0. It is called a lower point if T) < W and an upper point if T) > W. Let x be a unitary vector, with x ::::: ( 6 ) . We say that the nearly parallel line �x is downward if it contains an infinitely close lower point, and upward if it contains an infinitely close upper point.
Definition 7.3.18
=
As an example, consider N w0 x �£. w The line containing the vector l( fw ) is nearly parallel upward, and the line containing the vector ( l;w ) is =
nearly parallel downward. By convexity, a nearly parallel line cannot be both downward and upward with respect to a neutrix N. By the next proposition, if its intersection with N is not maximal, it should be either one.
Let N C �2 be a lengthy appropriately scaled and ori ented neutrix with length L. Let x ::::: ( 6 ) be a unitary vector. Assume that Tx < L. Then the nearly parallel line x is either downward or upward with respect to N. Proposition 7.3. 19
Proof. Let W be the width of N. Let A be unlimited such that Tx < A < L. By proposition 7.3.9 there exist orthonormal vectors u ::::: ( 6 ) and v ::::: ( ? ) such that N£l = £A EB Wv. By continuity x n £A EB 0 v contains some point y �u + T)V with � E L + and ITJI > W. If T) < W the line x is downward, and T) > W the line x is upward. D =
Let N c �2 be a lengthy appropriately scaled and oriented neutrix with length L and width W. Let tf. N be an infinitely close lower point and y tf. N be an infinitely close upper point, with llxll = II Y II · Let u ::::: ( 6 ) and v ::::: ( ? ) be orthonormal vectors such that u _l xy. If for some unlimited A with IAI � ll x ll it holds that Nn = £Au 8 Wv, the points x and y are called opposite with respect to N. Also, the lines �x and �y are called opposite with respect to N.
Definition 7.3.20
x
The final proposition of this section states that opposite lines generate the same thicknesses.
Let N C �2 be a lengthy appropriately scaled and ori ented neutrix. Let a, b ::::: ( 6 ) be unitary such that �a is a neaTly parallel downward line, and �b an opposite nearly parallel upward line. Then Ta Tb .
Proposition 7.3.21
=
7.3. Proof of the decomposition theorem
103
Proof. Let L be the length of N and W be its width. Let x E �a be an infinitely close lower point and y be its opposite infinitely close upper point on �b. Let A E L + , 1 > 1 ::::> llxll be such that N£>.. = £>u EB W where u ( 6 ) and v ::::: ( � ) are orthonormal vectors such that u _l xy. Let a > 0 be such that Ta < a < llxll . Then aa t/. N, so there exist � ::; ll x ll , T) > W such that aa = � u T)V . Because N£>.. = £>u EB W it holds that ab = � u + T)V tf. N, so a t/. Tb. Hence Tb C Ta . In a symmetric manner we prove that Ta C Tb . We conclude that Ta Tb . D v,
:::::
v,

=
On the division of neutrices
7.3.2
Let N C �2 be a neutrix with width W and length L. We assume that N is of the form N L ( 6 ) EB W ( � ) . Consider all lines �x with x of the form
x = (�), y E R
=
The following sets appear to be of interest:
{ y I � ( � ) n L ( 6 ) EB we ( � ) 0 } S = { y l �( � ) n Le ( 6 ) EB W ( � ) # 0 }
1. R 2.
=
=
(L ¥ �) .
Clearly, i f y E R or y E S the neutrix realizes its length in the direction x, i.e. one has Tx = L. If y E S. the line � ( � ) leaves N on its "small" side. On the other hand, if y E Re, the line �( � ) leaves N on its "large" side. But N may have "corners", i.e. , there may exist lines �x which after leaving N enter into the set Le ( 6 ) EB we ( � ) . Then S � R. An example of such a neutrix is N = £( 6 ) EB c:£ ( � ) , with 0, > 0. An example of a neutrix without such "corners" is given by N = £ ( 6 ) EB 0 ( n , see also [5] . If W ¥ L, the sets R and S are neutrices, in fact they result from an algebraic operation applied to W and L. The first one is the wellknown division operator on ideals or modules, commonly written :" [36] . We recall here its definition, in the context of neutrices. c :::::
c
"
Definition 7.3.22
Let M, N C � two neutrices. We write M:N
=
{ x E � I (Vn E N) ( nx E M) } .
One can also abbreviate by
M : N = {X E � I NX c M)} . Since N( £x) = (£N)x = Nx C M, the set M : N is a neutrix. Notice that
N(M : N)
c
M
104
7. Neutrices in more dimensions
and that lvl : N is the maximal set X which satisfies the property N · X C M.
(7. 1)
A s such, we call M : N the solution o f the equation (7. 1 ) . The second one is also a sort of division, that we note M/N . This division is based on a sort of inverse. Its definition needs more knowledge on neutrices, and will be postponed. The study of divisions is highly related, but distinct from earlier work on the division of ncutriccs by Koudjcti [29] (sec also [30] ) , mainly in the sense that our definitions are of algebraic or analytic nature, instead of settheoretic. We usc many of his tools and results. sometimes in a slightly modified form. Following Koudjeti, the argumentation becomes more simple and intrinsic, if instead of multiplications of neutrices we study additions of lower halftines. The transformation from neutrices N to lower halftines is done by the symmetrical logarithm log8 (N) = log(N + \ {0} ) ; formally we define log8 {0} = 0 . We transform halftines G back by the symmetrical exponential exp8( G ) = [ exp( G ), exp( G ) ] ; formally we define exp8 0 = {0}. Below we recall some fundamental properties of halftincs and ncutriccs. A neutrix I is idempotent if I · I = I. A lower halftine H is idempotent if H + H = H. A lower halftinc is idempotent if and only if it is of the form ( oc N) or ( oo, N] , where N is a neutrix. A lower halftine H is idempotent if and only if exp8 H is an idempotent neutrix. A neutrix N is idempotent if and only if log8 N is an idempotent lower halftine. Let c; ::: 0, c; > 0. Examples of idempotent neutrices are, in increasing order {0} ' £ e  @/c ' £ c; cfJ ' 0 ' £ ' £e ( l/c) ©J and � . The neutrices £ · e  ( l/c) 2  @ /c ' ( 1 /c: ) 1/c . £ . c;ciJ , ( 1 / c: ) · 0 , c:£ and £er ( l/c ) + ( l/c) @ are not idempotent . They are all of the form a I where I is an idempotent neutrix, and a is a real number. In fact every neutrix can be written is this form. This is a consequence of the following classification theorem of lower halflines, which with successive generalizations has been proved in [9, 3, 5] : 
,
·
·
·
Theorem 7.3.23 Every lower halfline H C � has a representation either of the form H = (  oo , r + N) or H = (  oo , r + N] , where N is a neutri:r:, which is unique, and r is a real number, determined up to the neutrix N . We see that a lower halfline H may b e written i n the form H = r + K , where K i s of the form (  x , N ) o r (  oo , N] , i.e. the lower halftine K is idempotent. Then exp8 H = e r exp8 K, and we obtained as a consequence that every neutrix is the product of a real number and an idempotent neutrix. With respect to the above theorem we recall some notation. Halftincs of the form (  oo , r + N) may be called open, and halftines of the form (  oo , r + N]
7.3. Proof of the decomposition theorem
105
closed. The external set r + N is called an external number; it has been shown [29] , [30] that many algebraic laws valid for the real number system continue to be valid for the external numbers. Certain analytic laws, too, on behalf of the above theorem. For instance, it may be justified to call the external number r + N the supremum sup H of H. We call H (  oo , sup H] the closure of H, and H ( oo. sup H) the interior of H. The pointwise addition of lower halflines H and K, satisfies =
=
sup H � sup K, or sup H = sup K and K is open (7.2) sup H � sup K, or sup H = sup K and K is closed.
Similar definitions and rules hold for upper halflines, working with infimums instead of supremums. Since lower halflines may be translated into idempo tent lower halflines, and neutrices may be rescaled to idempotent neutrices, in defining algebraic operations we may restrict ourselves to the idempotent cases. The extension to the general case is straightforward and will be briefly addressed to at the end of this section. We turn first to the problem of defining subtractions. The first opera tion ; will correspond to the division : , and the second operation . . , which will correspond to the division / , is defined through inverses. Definition 7.3.24
K by
Let H, K be idempotent lower halfiines. We define H : H : K = {x I (Vk E K)(k + x E H) } .
Notice that K + (H ; K) C H and that we have a maximality property similar to the division :, i.e. H ; K may be called the (maximal) solution of the equation K + X H. =
Definition 7.3.25 Let H be an idempotent lower halfiine. We define the sym metrical inverse ( H) s of H by
(H) s =
{ HH
H open H closed
If H is open ("boundary to the left of zero") , its symmetric reciprocal is a idempotent halfline, which is closed ("boundary to the right of zero") ; in a sense the "distance" of the "boundaries" of H and ( H8) to zero is equal. So there is some geometric justification in calling the reciprocal symmetric. Formally it holds that ( 0) s = � and ( �) s = 0 . Definition 7.3.26
Let H, K be idempotent lower halfiines. We define H . . K by H . . K = H + (K) s ·
106
7.
Neutrices in more dimensions
Notice that H . . K is an idempotent lower halftine, too. Proposition 7.3.27
sup K. Then 1 . ( H) s
Let H, K be lower halfiines. Let S = sup H and T =
= H0 .
2.
H . . K = H  K0 .
3.
H ·· K �
{
S � T, or S = T and H is closed S � T, or S = T and H is open.
:
(  )s
The proofs are straightforward. using formula (7.2) in 3.
Let H, K be two idempotent lower halflmes of lRL Let S = sup H and T = sup K. Then
Proposition 7.3.28
2.
H : K··
Proof.
{
S � T, or S = T and K is open S � T, or S = T and K is closed.
(  K) s H
1. Let x E H ; K. Suppose x E H0  K. Then there exist y > H, k E K such that x = y  k. Thus x + k > H, so x tf. H : K, a contradiction.
c,
c.
Then x E ( H0  K) hence H : K C ( H0  K) Conversely, let x E (H0  K) 0 . Suppose x tf. H : K. Then there exists k E K, z E H0 such that k + x = z . Thus x E H0  K, a contradiction. So x E (H0  K) 0 and (H0  K) 0 c H ; K. We conclude that H : K = (H0  K ( . 2. Straightforward, from 1 and formula (7.2).
D
The following theorem is a direct consequence of propositions 7.3.27.3 and 7.3.28.2. Theorem 7 .3.29
Let H, K be two idempotent lower halflines oflRL Whenever
H i= K, it holds that
but
H . . K = H : K,
7.3. Proof of the decomposition theorem
107
So, generically, the subtraction .. , defined through reciprocals, and the sub traction : , defined through solutions, yields the same result; thus the equation K + X = H can be solved through reciprocals. In the exceptional case of the subtraction of two identical halflines the outcomes are strongly interrelated, for H H is the interior of H ; H, and H ; H the closure of H H. We will now consider divisions. We start by defining the symmetric inverse. We relate this notion to the symmetric reciprocal. ··
· ·
Definition 7.3.30
Let N
C
inverse (N  1 ) 8 is defined by
� be an idempotent neutrix. The symmetric
Notice that (N  1 ) is an idempotent neutrix, for it is the exponential of an idempotent lower halfline. Below we calculate the symmetric inverse for some familiar neutrices. We have always c 0, c > 0. s
:::
{0}
£e  @/c £c: cfJ 0
0
�
(  oo , 0/c:) (  oo , £ log c:) (  oo , £) ( oo, £] (  oo , £/c:]
(  oo , 0/c:] ( oo, £ log 1/c:] ( oo, £] (  oo , £) (  oo , £jc:)
£ £e (l /c) Col � �
0
� £e 0/c
£ ( 1 /c:) @
£ 0
£e  cfJ/c
{0}
Table 7. 1 : Inverses of some idempotent neutrices. In [29] the symmetric inverse of a ( convex) set N is defined to be tfc U {0}. The next theorem states that, if N is a neutrix, the two definitions are equivalent. Proposition 7.3.31
Let N C � be a neutrix. Then ( N 1 ) s
=
1
Nc
U
{0}.
Proof. The equality holds formally for N = {0} and N = R Let 0 � N � R We prove only that (N  1 ) 8 C tfc U {0}. Clearly 0 E tfc U {0}. Now assume
108
7.
Neutrices in more dimensions
that x is nonzero element of (N  1 ) s, that we may suppose to be positive by reasons of symmetry. Then a
log x E (  logs N) s  (logs N)0  (log ( I N I \ {0})0  log ( � + \ N) log IR+\N . So x E 1 IN ° . Hence (N 1 ) s C Jc U {0}. D We define the symmetric division through multiplication by the symmetric inverse, and relate it to the symmetric subtraction. Definition 7.3.32 Let lvf, N C � be two idempotent neutrices. The symmet ric division MIN of M and N is defined by
Proposition 7.3.33
Let M, N be idempotent neutrices. Then
We have, using the algebraic relations M = exps logs M and exps H exps K = exps(H + K) , MIN = M (N 1 ) s Proof.
·
·
= exps logs M exps(  logs N) s = exps (logs M + (  logs N) s ) = exps (logs M . . logs N) . ·
D
We refrain from giving a general formula for MIN and point out that it can be calculated with the help of the propositions 7.3.27 and 7.3.33. However in some special cases MIN may be readily calculated. 1. M ¥ N c 0 : MIN = M. Nllv1 = (Ar 1 ) s . 2. £ c M ¥ N : MIN = (N 1 ) s , NIM = N. 3.
M c 0 : MIM
4. M ::) £ :
=
M.
MIM = (M  1 ) s .
7.3. Proof of the decomposition theorem
109
Observe that always MjM C 0. We consider now the division M N, of definition 7.3.22. It bears the following relation to the subtraction : : Proposition 7.3.34
Let M, N be idempotent neutrices. Then :
We prove only that M N C exp s (logs M ; logs N) . Formally, one has 0 E exps (logs AI : logs N) . Let x E .M : N, x # 0. For reasons of symmetry, we may suppose that x > 0. Because xN C lvl one has log x + logs N C logs M . S o log x E logs lvl ; logs N, hence x E exps(logs M ; logs N) . We conclude that M : N C exps (logs M : logs N) . D Proof.
As a consequence of theorem 7.3.29 and propositions 7.3.33 and 7.3.34 we obtain that M/N M N whenever M # N. If M N we have =
:
=
Mc0 M ::) £.
Notice that always M : M ::) £. Because Mjlvl C 0, we obtain MjM � M : M.
The division has the following settheoretic characterization. Proposition 7.3.35
Let M, N C � be two neutrices. Then M:N=
(
Me
N \ {0}
)c
The proof is very similar to the proof of proposition 7. 3. 28 . 1 . D It follows from the results above that, analogously to the subtraction : , the division fiv1 : N can in practice be calculated through inverses. Sec also the table and the special cases we presented earlier. We indicate briefly how subtractions of nonidempotent halftines can be reduced to subtractions of idempotentent halftines, and consider also the anal ogous reduction for divisions of neutrices. Let F, G be two lower halftines. By theorem 7.3.23 there exist real numbers f and g and idempotent halftines H and K such that Proof.
F=f+H
G = g + K.
110
7.
Neutrices in more dimensions
We define
F . . G = f  g + H .. K
F : G = f  g + H : K.
In the same manner, let AI, N be neutrices. Let potent neutrices such that
M = mi
m,
n E � and I, J be idem
N = nJ.
We define
MjN = rn n IjJ. As regards to the operation M : N, the relation m M:N=( I J) n :
(7.3)
readily follows from definition 7.3.22. It is a matter of straightforward verifi cation to show that the above formulae do not depend on the choice of f, g, m and n, and that, mutatis mutandis, the properties considered earlier in this section continue to hold. We state three useful properties of the division : . We recall that a neutrix N is linear if there exists c; > 0 such that N = 0c: or N = £c:, else N is nonlinear. Nonlinear neutrices have the property that N = w N for at least some w "'=' +oo ( see [5] ) . ·
Let AI, M1 , l'vi2 , N, N1 , N2 C � be neutrices. If l'vh c M2 , it holds that M1 : N c M2 : N. If N1 c N2 , it holds that M : N1 ::) M : N2 . If M � N, one has M : N = 0 if and only if there exists c > 0 such that M 0c and N £c:, otherwise M : N � 0 .
Proposition 7.3.36 1. 2. 3.
=
=
We only prove 3. Let c:, T) > 0. As regards to linear neutrices we have the following table
Proof.
0c: £c: 0T) £c:!TJ 0c/TJ £T) £TJ/c £T)jc So the only possibility for such neutrices !vi, N to obtain M : N 0, is when M = 0c and N = £T), with c/T) appreciable, i.e., when N = £c:. Else the condition AI � N ensures that c/TJ 0, respectively TJ/c 0, which implies that M : N � 0. =
"'='
"'='
7.3. Proof of the decomposition theorem
111
Assume M is nonlinear. Let w ::: +oo be such that
M:N
=
A1 :N w
1 ( M : N)
= 
w
1
c 0
w
Mlw =
A1 .
Then
� 0.
The case that N is nonlinear is similar. This concludes the proof. D Finally we establish the relation between the two divisions and the families of directions in the plane R and S. Theorem 7.3.37 Let N c form N = Lu EB W v, where
�2 be a neutrix with width W and length L, of the u and v are orthonormal vectors. Let
and, if L ¥ �� let Then R = W : L and S = WIL . Without restriction of generality, we may assume that u ( 6 ) and v = ( � ) . First, let y E R. If y = 0, clearly, y E W : L. Assume y i= 0. Then there exist A E L, A f. 0 such that ( }y ) tt L ( 6 ) EB we ( � ) . so =
Proof.
(
)
AY we e = W : L. y=E L \ {0} A =
Hence R C W : L. Conversely, let y E W : L. If y 0, clearly y E R. Assume y f. 0. Suppose there is A E L such that ( .\.\y ) E L ( 6 ) EB we ( � ) . Then y E L\{�} , which means that y tt W : L, a contradiction. So y E R, hence W : L C R. We conclude that R W : L. Second, let y E S. Then there exists p, E Le such that ( /}y ) E Le ( 6 ) EB W ( � ) . So p,y E W and y E � = WI L. Hence S C WI L. Conversely, let y E WI L. Then there is p, E Le and T) E W such that y = '111 . So ( /}y ) E Le( b ) EB W 0 ) , which means that y E S. Hence WIL c S. We conclude that S = WI L. D =
7.3.3
Proof of the decomposition theorem
Let us consider a neutrix in �2 . We know already that, if it is square or not lengthy, it can be decomposed into two neutrices of R For lengthy neutrices, we sketch here the remaining part of the proof of the decomposition theorem.
112
7.
Neutrices in more dimensions
A lengthy neutrix may be assumed to be appropriately scaled and oriented. By theorem 7.3.8 it suffices to look for a direction in the plane with maximal thickness. The set of directions with maximal thickness will be obtained as the complement of the directions with nonmaxirnal thickness, where by the sectortheorem we may restrain ourselves to directions nearly parallel to the ncutrix. By proposition 7. 3 . 1 9 and 7.3.21 they arc divided into two "equal" opposite parts. The set of directions with maximal thicknesses will then be the supremum, in the sense of theorem 7.3.23, of the set of downward nearly parallel directions, or alternatively, the infimum of the set of upward nearly parallel directions. In fact, the "gap" between the two opposite families of directions is of the form ( x +�:L ) , where W is the width of the neutrix, and L its length. We turn now to the proper proof, and start with some terminology. Definition 7.3.38
neu.trix. We write D U SD Su For y ::: 0 we write
Let N C �2 be a lengthy appropriately scaled and oriented :::
{ y 0 I ( � ) is nearly parallel downward } { y ::: 0 I ( � ) is nearly parallel upward } { x E �I D + x = D } { x E �1 U + x = U } .
and we define I=
n Iy.
yeo: O
Let N C �2 be a lengthy appropriately scaled and oriented neutrix with length L and width W . Then
Theorem 7.3 .39
I = W : L � 0.
It follows from proposition 7.3.36.3 that W : L � 0. Let x E I. Suppose X tt w : L. By proposition 7. 3.35 there are T) E w e and A E L such that x = TJ/ A. By proposition 7.3.9 there arc orthonormal vectors u ::: ( 6 ) and v ::: ( � ) such that N£>.. = £Au EB W v; up to a rotation we may assume that u ( 6 ) and v ( � ) . So ( � ) tf. N . Hence TG ) c T( )' / >. ) ¥ £A c T( 6 ) . Hence x t/. Iu ::) I, a contradiction. This implies that x E W : L. which means that I c W : L. Conversely, let x E W : L. Let y ::: 0. By proposition 7. 3 . 1 1 there are orthonormal vectors u , v such that Nr(t) = T ( � ) u EB Wv. It follows from Proof.
=
=
7.3. Proof of the decomposition theorem
113
theorems 7.3.37 and 7.3.8 that T( y �x) T( t ) for all x E W : T( ; ) ::) W : L, so x E Iy . Because y is arbitrary, it holds that x E I. Hence W : L C I. We conclude that I = W : L. D =
Let N C �2 be a lengthy appropriately scaled and oriented neutrix with length L and width W. Then 1 . For all y, z E D such that y ::; z one has T(�) ::; TG) .
Theorem 7.3 .40
2.
For all y, z E U such that y ::; z one has TG ) z
:2
TG) .
For all c > W : L there is y E D, E U such that z  y ::; c. 4. SD = Su = W : L. 5. There exists x ::: 0 such that D = [0, x+ W : L ) and U = (x + W : L, 0] . 3.
Proof.
1 . Suppose T(t) > TG) . By proposition 7.3.21 there is x E U such that T( ! ) = T( � ) . So y < z < x, y ::: x , while T( � ) > T( ! ) < T( ; ) . This contradicts the sectortheorem. Hence TG ) ::; TG) .
2. Analogous to 1 . 3 . Because W : L is a neutrix, one has c/2 > W : L. By proposition 7.3.35 there exist A E L, T) > W such that TJ/ A ::; c/2. Let v be orthonormal such that N£>.. £Au EB W . We see that AU + T)V, AU  T)V tf. N, so  TJ/A E D and TJ/A E U, while TJ/A  (  TJ/A) ::; c. 4. From 3 we derive that SD C W : L. Conversely, let y E D. Then it follows from theorem 7.3.39 that T( y+W1 ) = T(1) y < L. This implies that y + x E D for all x E W : L. Hence W : L C SD . We conclude that SD = W : L. The proof that Su = W : L is analogous. 5. By theorem 7.3.23 and 4 the set D is either of the form D = [0 , x+W : L ) or D [0, x + W : L ] . We show that the second possibility is absurd. Then the only way to satisfy 3 is when U (x + W : L, 0] . By 7.3.39 and 1 one has T( � ) ::; T( ! ) for all y E D. Then also TG) ::; T(�) for all z E U. by proposition 7. 3.21. Because D U U = 0, one has TC ) ::; TG) for all y ::: 0. By theorem 7.3.6 all other thicknesses are equal to W. Hence Tr ::; T(;) < L for all unitary vectors r. Then L cannot be the length of N, a contradiction. Hence D = [0 , x + W : L ) . The proof that U (x + W : L, 0] is analogous. D =
11,
v
L
=
=
=
114
7.
Neutrices in more dimensions
Let N C �2 be a lengthy appropriately scaled and oriented neutrix with length and width W. Then there exists x � 0 such that y I T( �) = = X + w
Theorem 7.3.41
L}
{
L
L.
:
By theorem 7.3.40.5 there exists x � 0 such that 0 \ (D U U) = By theorem 7.3.39 we have T( �) = TG ) for all y E x + W x+W Suppose TG ) < By proposition 7.3. 19 either x E D or x E U, a contradic tion. Hence T( ! ) ::::> in fact T(;) for all y E x + W D
Proof.
:
L.
L.
:
L,
L
=
Theorem 7.3 .42 ( Twodimensional
a neutrix. Then there are neutrices vectors u, v such that N
=
:
L.
�2 be C �� and orthonormal
decomposition theorem) Let N
L, W with W C L Lu Wv .
L.
C
EB
L is the length of N, and W is its width. Proof. Let L be the length of N and W its width. If L W one has N = L( 6) W ( � ) . If L = £>. for some A E �' the theorem follows from Moreover, the neutrix
=
EB
proposition 7.3.9. In the remaining cases we may assume that N is appropri ately scaled and oriented. By theorem 7.3.41 there exist a unitary vector u such that Tu Let v be unitary such that u, v are orthonormal. By the orem 7.3.8 we have N = u EB Wv . The last part of the theorem also follows from theorem 7.3.8. D =
L.
L
References
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VAN DEN BERG,
Nonstandard Asymptotic Analysis, Lecture Notes in
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VAN DEN BERG , An external utility function, with an application to mathematical finance, in H. BacelarNicolau, ed., Proceedings 32nd European Conference on Mathematical Psychology, National Institute of Statistics INE. Portugal (2001).
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BoHE, "The existence of supersensitive boundaryvalue problems", Methods Appl. Anal., 3 ( 1996) 318334.
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BoSGIRAUD, "Exemple de test statistique non standard", Ann. Math. Blaise Pascal, 4 ( 1997) 913.
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J BoSGIRAUD, "Exernple de test statistique non standard, risques ex ternes", Publ. Inst. Statist. Univ. Paris, 41 (1997) 8595. .
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Nonstandard likely ratio test in exponential families, this
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Asymptotic Analysis, North Holland, 1961.
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Nonstandard metho ds for additive and combinatorial number theory. A survey Renling Jin *
8.1
The beginning
In this article my research on the subject described in the title is summa rized. I am not. the only person who has worked on this subject. For example, several interesting articles by Steve Leth [21 , 22, 23] were published around 1988. I would like to apologize to the reader that no efforts have been made by the author to include other people's Tesearch. My research on nonstandard analysis started when I was a graduate student in the University of ·wisconsin. A large part of my thesis was devoted towards solving the problems posed in [1 9] . By the time when my thesis was finished, many of the problems had been solved. However, some of them were still open including [19, Problem 9 .1 3 [ . It took me another three years to find a solution to [ 19, Problem 9. 13] . Before this my research on nonstandard analysis was mainly focused on foundational issues conceming the structures of nonstandaJ:d universes. After I told Steve about my solution to [ 1 9 , Problem 9 . 1 3] , he immediately informed me how it could be applied to obtain interesting results in combinatorial number theory. This opened a stargate in front of me and lead me into a new and interesting field. For nonstandard analysis we use a superstructme approach. We fix an �1saturated nonstandard universe * V . For each standard set A we write *A for t.he nonstandard version of A in *V. *
Department of JV[athematics, College of Charleston, Charleston, SC 2942 (k + k'  2)d, A = G + gN, and B = G' + gN. Then 4(A + B) = Roughly speaking, Kncscr's Theorem = 4(A) + 4(B) says that the only kind of counterexamples which make the inequality false in Mann's Theorem with J replaced by 4 arc similar to the one just described. In [13] a parallel theorem [13, Theorem 3.8] was obtained. Let A, B � N with BD (A) = a and BD (B) = (3. Then there are intervals [a n , bn ] and [en , dn] " n> oc " nHXl (bn  a n ) " n>CXl (d n  Cn ) sue·h th at l lm  oo, l lm  oo , l lm
k+;l
�
n[aan,n bnll l IAbn+
���_[c�!n}l
and limn___, oc = (3. We hope to characterize the structure of A + B inside the intervals [an + en , bn + dn] . However, [13, Theorem 3.8] when restricted to the addition of two sets, only characterized the structure of A + B on a very small part of N. The reason for this is because we used only one x and one y with 4((*A  x ) n N) = a and 4((*B  y) n N) = (3 in the proof. During the summer of 2003 my undergraduate research partner Prerna Bi hani and I conducted an undergraduate research project funded by the College of Charleston to work on theorems parallel to Kneser's Theorem. The work done during the summer and the following year produced the paper [2] , which contains Theorem 8.4. 1 . To avoid some technical difficulties we considered only the sum of two copies of the same set in [2] . a,
4
The version of Kncser's Theorem in
[9]
is about the addition of multiple sets. We stated
the version here for the addition of two sets just for simplicity.
8.5.
Inverse problem for upper asymptotic density
1 25
Theorem 8.4. 1 Let A be a set of nonnegative integers such that BD (A) = a and BD (A + A) < 2 a . Let {[an, bn ] : n E N} be a sequence of intervals such . ,b, ] l �T " that llmn_, = (b n  an )  00 an d l lmn_, = IAn[a, bn  an+ l  a . Th en th ere are g E n , G c;;;; [0, g  1], and [en , dn ] c;;;; [an , bn ] for each n E N such that . (1) llmn> CXJ dnCn bn  a n  1 _
'
(2) A + A c;;;; G + gN, (3) (A + A) n
[2en , 2dn ]
(4) BD (A + A) = JQl g
= (G + gN) n
[2cn, 2dn ] for all n E N,
a
? 2  lg .
Nate that ( 1) above shows that the structure of A + A is characterized on a large portion of [2a n , 2bn ] . Note also that we cannot replace [2cn , 2dn] with [2an , 2bn ] in (3) because all conditions for A still hold if we delete any elements from A n ( [an, bn ] "' [en, dn ] ) . The proof of Theorem 8.4. 1 can b e described i n several steps. Given a hyperfinite integer N. we know that for almost all x, y E [a N , b N ] we have g((*A x) nN) = g( (*A  y) nN) = a . We can also assume that g((x  *A) nN) = g((y  *A) n N) = a . Step one: characterize the structure of *(A + A) in x + y + Z using Kneser's Theorem, where Z is the set of all standard integers. Step two: show that the structures of *(A+A) n (x +y+Z) for almost all x, y E [a N , bN ] are consistent with one another so that these structures can be combined into one structure. Hence we can characterize the structure of *(A + A) in [2cN , 2dN] , � 1 . Step three: prove that for different hyperfinite integers N where and N', the structure of *(A + A) in [2a N, 2bN ] and the structure of *(A + A) in [2a N' ' 2bN' ] are consistent so that these structures of *(A + A) in [2a N , 2b N ] for all hyperfinite integers N can be combined into one structure of *(A + A) in U{[a N , bN ] : N is hyperfinite } . Step five : pushing down the structure of *(A + A) to the standard world results Theorem 8.4. 1 . The methods developed in [13] do not seem to b e enough for proving The orem 8.4. 1. So it is interesting to see whether one can produce a reasonably nice and short standard proof of the theorem. In [3] the structure of A was characterized when g(A) is very small and g(A + A) "( cg(A) for some constant c ? 2. It is also interesting to see how one can characterize the structure of A + A when BD (A + A) "( cBD (A) for some constant c ? 2.
�z=��
8.5
Inverse problem for upper asymptotic density
In January of 2000, I was invited to give a talk at the DIJ\IACS workshop "Unusual Applications of Number Theory". One of the workshop organizers
8.
126
Additive and combinatorial number theory
was Melvyn Nathanson to whom I am grateful for being the first number theorist to express an interest in my research on number theory not to mention his continued encouragement. During the workshop I had a chance to meet another number theorist G. A. Freiman who is wellknown for his work on inverse problems in additive and combinatorial number theory. He gave me a preprint of his list of open problems [5] . This list and the book [24] have since gotten me interested in the inverse problems. Inverse problems study the properties of A when A+ A satisfies certain con ditions. Freiman discovered a phenomenon that if A + A is small, then A must have some arithmetic structure. In fact Kneser's Theorem and Theorem 8.4.1 can b e viewed a s two examples o f the phenomenon. One can characterize the arithmetic structure of A from the structure of A + A in Theorem 8.4. 1 and characterize the structure of A and the structure of B from the structure of A + B in Kneser's Theorem (see [2] for details ) . In this section we characterize the structure of A when the upper asymptotic density of A + A is small. Given A � N, the upper asymptotic density d(A) of A is defined by A n [1, n] l d( A ) = lim sup l _ n ':' <Xl
n
Without loss of generality we always assume 0 E A in this section. We can also assume that gcd(A) = 1 because if gcd(A) = d > I , then we can recover the structure of A from the structure of A', where A' = { a/ d : a E A } . When 0 E A and gcd(A) = I , one can easily prove, using Freiman's result ( 1 ) at the beginning of the next section, that d( A + A) ? � d( A) if d( A) :'( � and
d(A + A) ? l +�( A ) if d(A) ? �  The following two examples show that the
lower bounds above are optimal. Example 8 . 5 . 1
For every real number 0 :'( a :'( 1, let 00
{0} U U [1(1  a)22nl , 2 2n ] . n= l Then d(A) = a, d(A + A) = 11" if a ? �� and d(A + A) = �a if a :'( � A=
Let k , m E N be such that k ? 4 and 0 , m , 2 m are pairwise distinct modulo k. Let A = kN U (m + kN) . Then d(A) = � = a :'( � and d(A + A) = t = �a . It is easy to choose k, m such that gcd (A) = 1 . S o we can say that d(A + A) is small whm d(A + A) = min g J(A), l+�( A ) }
Example 8.5.2
and we need to characterize the structure of A when d( A + A) is small. Clearly the characterization of the structure of A should cover the cases in both Example 8.5.1 and Example 8.5.2. We hope to show that A must have
8.5. Inverse problem for upper asymptotic density
127
the structure described in one of the examples above when d(A + A) is small. However, some variations of the examples are unavoidable. If the set A is replaced by A' 0.
Part 1: Assume a > �  Then d(A + A) = 11a implies that for every increasing I An[O,+hn] l = a, we h ave · sequence { h n : n E nRT } wz·th 1 1m17_,00 hn l nlim >oo
I (A + A) n [0, h n ] l hn + 1
= a.
Part 11: Assume a < � and gcd(A) = 1 . Then d(A + A) = �a implies that either (a) there exist k > 4 and c E [ 1 , k  1] such that a = f and A oo hn and [ n + 1 , bn  1] n A = 0 for every n E N. 
e
8. Additive and combinatorial number theory
128
Part III: Assume a = � and gcd (A) = 1 . Then d( A + A ) = � a implies that ei ther (a) there exists c E { 1 , 3} such that A � 4NU (c+4N) or (b) for every increasing sequence { hn E N } with limn >exl I A�lo+�"l l = a , we have :
n
nlim > oo
I (A + A ) n
hn
[0, h n ] l = +1
a.
I would like to make some remarks here on Theorem 8.5.3. First, the proof of Part I is easy; the most difficult part is Part II. Second, Part I and (b) of Part III cannot be improved so that set A has the structure similar to the structure described in (b) of Part II. For example, if one lets A = {0} 
U

00
u ( [3 l
n=
•
22n 3 ' 4 22n 3 ] U [5 22n 3 ' 22n ]) ' •
•
l +d(A)
then d ( A ) = 21 and d ( A + A ) =  . Clearly A does not have the structure 2 described in (b) of Part II. The main ingredient of the proof of Theorem 8.5.3 is the following lemma in nonstandard analysis. For an internal set A � [0, and a cut U � [0, we define the lower Udensity du (A) by
{ { CA�lO�n] l ) : n
du (A) = sup inf st where
g(A)
H]
E U " [O, m]
}
}
:mEU ,
means the standard part map. Note that if U N and A � N, then du (*A). A set I = { a , a + d, a + 2d, . . . } is called an arithmetic progres
st
=
H]
=
sion with difference d. An arithmetic progression can be finite (hyperfinite) or infinite. If an arithmetic progression is finite (hyperfinite) , then its cardinality (internal cardinality) is its length. A set I U J is called a hiarithmetic pro gression if both I and J are arithmetic progressions with the same difference d and I + I, I + J, and J + J are pairwise disjoint. A finite (hyperfinite) hiarithmetic progression I U J has its length I I I + I J I . Let U be a cut. A hiarithmetic progression B � U is called Uunbounded if both I and J are upper unbounded in U.
H
O *].
H]
Lemma 8.5.4 Let be hyperfinite and U = nn E N [ , Suppose A � [0, be such that 0 < du (A) = a < �. If A n U is neither a subset of an arithmetic progression of difference greater than 1 nor a subset of a U unbounded bi arithmetic progression, then there is a standard positive real number 1 > 0 such that for every N > U, there is a K E A, U < K < N, such that
I (A + A ) n [o, 2K J I 2K + 1
[o, K
lA n JI 3 � 2K + 1
+
I·
8.6. Freiman's
3k  3 + b conjecture
1 29
Lemma 8.5.4 is motivated by Kneser ' s Theorem. It basically says that either A + A is large in an interval [0, 2K] with K > {f for some standard n or A has desired arithmetic structure in an interval [0, K] with K > {f for some standard n. The proof uses the fact that U is an additive semigroup. This can be done only in a nonstandard setting. It is interesting to see whether this lemma can be replaced by a standard argument with a reasonable length. Recently G. Bordes [4] generalized Part II of Theorem 8.5.3 for sets A with small upper asymptotic density. He characterized the structure of A when d(A) :( ao for some small positive number ao and d(A + A) < i J(A). It is interesting to see whether one can replace ao by a relatively large value, say � , in Bordes' Theorem.
Freiman's 3k  3 + b conjecture
8.6
After Theorem 8.5.3 was proven, I realized that the same methods used there could also be used to advance the existing results towards the solution of Freiman's 3k  3 + b conjecture [5] . This is important because the conjecture is about the inverse problem for the addition of finite sets. Let A be a finite set of integers with cardinality k > 0. It is easy to see that lA + AI � 2k  1. On the other hand, if lA + A I = 2 k  1 , then A must be an arithmetic progression. In the early 1960s, Freiman obtained the following generalizations [6] . ( 1 ) Let A 3 and I2AI = 2 k  1 + b < 3k arithmetic progression of length at most
(3)
 3, then k+b
A is a subset of an
If k > 6 and I2AI = 3k  3, then either A is a subset of an arithmetic progression of length at most 2k  1 or A is a hiarithmetic progression.
In [6] a result was also mentioned without proofs for characterizing the structure of A when k > 10 and lA + AI = 3k  2. In [10] an interesting generalization of (3) above was obtained by Hamidoune and Plagne, where the condition I2AI = 3k  3 is replaced by lA + tAl = 3k  3 for every integer t. However, no further progress of this kind had been made for a larger value of lA + A I before my recent work. In fact, Freiman made the following conjecture in [5] five years ago.
Conjecture 8.6.1 There set of integers A with I A I
exists a nat·ural rwmber K S1L Ch that for any finite k > K and l A + A I 3k  3 + b < 13° k  5 for
=
=
8. Additive and combinatorial number theory
130
some b ? 0, A is either a subset of an arithmetic progression of length at most 2k  1 + 2b or a subset of a biarithmetic progression of length at most k + b. Note that the conclusion of Conjecture 8.6. 1 could be false if one allows 1 l A + A I = 3° k  5. Simply let A be the union of three intervals [O, a  1 ] , [b, b+ a  1] , and [2b, 2b+a  1 ] , where k = 3 a and b is a sufficiently large integer. 1 Clearly lA + A I = 3° k  5. Since b can be as large as we want , we can choose a b so that set A is neither a subset of an arithmetic progression of a restricted length nor a subset of a hiarithmetic progression of a restricted length. Using nonstandard methods such as Lemma 8.5.4, I was able to prove the following theorem in [ 1 7] .
Suppose f N N is a function with limn_,oc f�n) = 0. There exists a natural number K such that for any finite set of integers A with I A I = k, if k > K and lA + AI = 3k  3 + b for some 0 :'( b :'( f(k), then A is either a subset of an arithmetic progression of length at most 2k  1 + 2b or a subset of a biarithmetic progression of length at most k + b. Theorem 8.6.2
:
c+
Theorem 8.6.2 gives a new result even for f(x) = 2. However, we still have a long way before solving Conjecture 8 . 6 . 1 . It is already interesting to sec whether we can obtain the same result with f(x) = ax for some positive real number a. The ideas for proving Theorem 8.6.2 are similar to the proof of Theo rem 8.5.3, but much more technical. Suppose Theorem 8.6.2 is not true: then one can find a sequence of counterexamples An such that I An I c+ oo . Given a hyperfinite integer N, let A = AN . Without loss of generality, we can assume that 0 = min A, H = max A, gcd (A) = 1. and a � 1 » 0. Note that
J!
I A+N�/'A I+3 � 0.
Hence l A + AI is almost the same as 3 I A I  3 from the non standard point of view. Using the casebycase argument, we can show that if 1 « � , then A is a subset of a hiarithmetic progression. If 1 � � and b = l A + A I  3 I A I + 3, then we can show that H + 1 :'( 2 I A I  1 + 2b when A is not a subset of a hiarithmetic progression. The proof for the case 1�!
J!
J!
J!
J!
is much harder than the proof for the case 1 « � although the former depends on the latter. In both cases Lemma 8.5.4 was used to get structural information of A on an interval with length longer than !i n for some standard positive integer n. There are some similarities between our methods and analytic methods. In order to detect some structural properties of A U is significantly greater than I A I / H , which will lead t o a contradiction that l A + AI i s almost the same as 3k  3. If du (A) = 0, then the density of A on [K, H] is significantly greater than I A I / H , which will again lead t o a contradiction. Otherwise either I (A + A) n [ 0 , 2K] I � � a, or A has very nice is large, which is impossible by the fact that structural properties on [0, K] following Lemma 8.5.4, which will force A to have the structure we hope for.
�'jt:11
References [1] V. BERGELSON, Ergodic Ramsey theoryan update, in Ergodic theory of zd actions (Warwick, 19931994), London Mathematical Society Lecture Note Ser. 228, Cambridge University Press, Cambridge, 1996. [2] P. BIHANI and R. JIN, Kneser's Theorem for upper Banach density, sub mitted. [3]
BI LU , "Addition of sets of integers of positive density", Journal of Number Theory, 64 (1997) 233275.
Y.
[4] G. BORDES , "Sumsets of small upper density", Acta Arithmetica, to appear. [5] G . A . F REIMAN, Structure theory of set addition. II. Results and prob lems, in Paul Erdos and his mathematics, I (Budapest, 1999 ) , Bolyai Soc. Math. Stud . , 1 1 , Janos Bolyai Math. Soc . , Budapest , 2002 . [6] G . A . FREIMAN, Foundations of a structural theory of set addition, Trans lated from the Russian. Translations of Mathematical Monographs, Vol. 37, American Mathematical Society, Providence, R. I., 1973. [7] H. FURSTENBERG , Recurrence in Ergodic Theory Nu.mber Theory, Princeton University Press, 198 1 .
and Combinatorial
[8] T . C oWERS, "A new proof o f Szemercdi 's theorem", Geometric and Func tional Analysis, 1 1 (2001) 465588. [9] H. H ALBERSTAM and K . F . ROT H , 1966.
Sequences,
Oxford University Press,
8. Additive and combinatorial number theory
132
[10] Y. 0 . HAJ\IIDOUNE and A . PLAGNE, "A generalization of Freiman's 3k  3 theorem", Acta Arith. , 103 (200 2 ) 147156. [ 1 1] R . .TIN, "Sumset phenomenon", Proceedings of American Mathematical Society. 130 (2002) 855861 . [12] R . JIN, "Nonstandard methods for upper Banach density problems·· , The Journal of Number Theory, 91 (2001) 2038. [13] R. JIN, Standardizing nonstandard methods for upper Banach density problems, in the DIMACS series, Unusual Applications of Number Theory, edited by M . Nathanson, Vol. 64, 2004. [14] R. JIN, "Inverse problem for upper asymptotic density", Transactions of American Mathematical Society, 355 (2003) 5778. [ 1 5] R. JIN, "Inverse problem for upper asymptotic density II", to appear. [16] R . .TIN. "Solution to the inverse problem for upper asymptotic density", to appear. [ 1 7] R. JIN, Freiman's 3k3+b conj ecture and nonstandard methods, preprint. [18] R. JIN and H . J . KEISLER, "Abelian group with layered tiles and the sumset phenomenon", Transactions of American Mathematical Society, 355 (2003) 7997. [19] H . J . KEISLER and S . LETH, "Meager sets on the hyperfinite time line", Journal of Symbolic Logic, 56 ( 1 99 1 ) 71102. [20] A. I . K H INC H IN , Three pearls of number theory, Translated from the 2d ( 1 948) rev. Russian ed. by F. Bagemihl, H. Komm, and W. Seidel, Rochester, N.Y. , Graylock Press, 1952. [2 1] S . C. LETH . "Applications of nonstandard models and Lebesgue measure to sequences of natural numbers", Transactions of American Mathematical Society. 307 ( 1988) 457468. [22] S . C . LETH, "Sequences in countable nonstandard models of the natural numbers", Studia Logica, 47 ( 1988) 243263. [23] S . C . LETH , "Some nonstandard methods in combinatorial number the ory", Studia Logica. 47 ( 1 988) 265278.
Additive Number TheoryInverse Problems and the Geometry of Sumsets, SpringerVerlag, 1996.
[24] M . B . NATHANSON,
Nonstandard methods and the Erdos Turan conj ecture Steven C. Leth *
9.1
Introduction
A major open question
in combinatorial
conj ecture which states that if with the property that arithmetic progressions
A
2:: �=1 1/an
[1[.
=
number theory is the ErdosTunin
(a71 ) is a sequence of natural numbers
diverges then
A
contains arbitrarily long
The difficulty of this problem is underscored by the
fact that a positive ans·wer would generalize Szcmer6di's theorem which says that if a sequence
A
C
N has positive upper Banach Density then
A
contains
arbitrarily long arithmetic prog;:ressions. Szemeredi's theorem itself has been
object of int en se int;crcst since first, conject.urocl, al so by Erdos and Turan , 1936. First proved by Szomerf)di in 1974 [91 , the theorem has been reproved using completely different approaches by Fnrstcnherg in 1977 [2, 3[ and Gowers in 1999 [4[ , with each proof introducing powerful new methods. the
in
The ErdosTuran conjecture immediately implies that the primes contain
arbitrarily long arithmetic progressions, and it was thought by many that a
successful proof for the pri mes would be the result of either a proof of the conj ectme itself re
c en tly
Green
or
significant progress toward the conj ecture. However, very
and Tao were able to solve the question for the primes without
generalizing Szemeredi's result in terms of providing weaker density conditions on a sequence guaranteeing that it contain arithmetic progressions.
In this paper we outline some possible ways in which nonstandm·d methods
might be able to provide new approaches to attacking the Erdos Tun'tn conjec ture, or at least other questions about the existence of arithmet.ic progressions. Heavy reference will be made to result:s iu
[7[ �:tnd [8[ ,
and the proofs for all
results quoted but not proved here appear in those two sources. *
Department of Mathematical Sciences,
co 80639.
steven . l
[email protected] o . edu
University of Northern Colorado,
Greeley,
9.
134
9.2
Nonstandard methods and the ErdosTun1n conjecture
Near arithmetic progressions
We begin with some definitions that first appear in
[8] .
Let A C N, and let I = [a, b] be an interval in N. We will write l (I) for the length of I (i. e. l (I) = b  a + 1) and we will write 8 (A, I) or 8(A, [a, b] ) for the density of the set A on the interval I. Th1LS 6(A, I) = ���{1 .
Definition 9 . 2 . 1
Definition 9.2.2 Let t, d and w be in N , and let a E � with 0 < a < 1. For A C N and I an interval in N of length l (I) we say that A contains a ttermed ahomogeneous cell of distance d and width w in I or simply a < t, d, > cell in I iff there exists b E I with b + (t  1)d + w also in I such that for each v, � = 0, 1 , 2, . . . ' t  1 : 6 (A, [b +�· d, b +�· d + w l) :::0: ( 1 a) c5 ( A , [b + v · d, b + v · d + w l) :::0: (1 a) 2 c5 (A, I). o:,
w
If each 6 (A, [b + .; · d, b + � · d + w] ) is simply nonzero, i. e. the intervals are nonempty, then we say that A contains a < t, d, > cell. For (3 > 0 and 0 ::; u ::; w we will say that a < t, a, d, w > cell is u, (3 uniform if for each v = 0. 1 , 2, ... , t  1, and all x such that u ::; x ::; w : w
( 1  {3 )6(A, Jv ) ::; cl (A, [b + V · d , b + V · d + x] ) ::; ( 1 + {3 )6(A , Jv ) · where Jv denotes the interval [b + v · d, b + v · d + w] . It is clear that an actual arithmetic progression of length t and distance is an example of a < t, a, w, d > cell with w = 0 and a any nonnegative number. Furthermore, this cell is u, (3 uniform for u = 0 and any nonnegative (3. We could view the existence of a < t, a, d, w > cell inside a sequence A as a weak form of an actual arithmetic progression inside A . These cells arc "ncar" arithmetic progressions in some (perhaps rather weak) sense, and intuitively arc "nearer" to arithmetic progressions as the size of w decreases. In some of the results that we look at w will be "small" in the sense that the ratio of w to d will be small compared to the ratio of d to the length of the interval I. In other results w will be "small" by actually being bounded by a finite number while d gets arbitrarily large.
d
Definition 9.2.3 Let I be an interval in N, and A C I, with r > 1 E � and m E N. We say that A has the m, r density property on I iff for any interval
J C I, if l(J)
:::0:
l�) ,
then 6(A, J) ::; r6(A, I) .
Theorem 9.2.1 below gives a condition for the existence of "ncar" arithmetic progressions for any sequence on any interval I in which the density does not
9.2.
Near arithmetic progressions
135
drastically increase as the size of the subinterval decreases. More specifically, it provides an absolute constant such that whenever the density of a sequence does not increase beyond a fixed ratio for any subinterval of size greater than the length of I divided by that fixed constant, then the sequence will contain a < t, a, w. d > cell with some relative "smallness" conditions for w. A complete proof o f this theorem appears i n [8] , but we will outline the proof here. as it provides the clearest illustration of how the use of the non standard model provides us with a new set of tools for questions of this type.
Theorem 9 . 2 . 1 Let h(x) be any increasing h(x) > 0 whenever x > 0, and let g(x) be
real valued function such that any real valued function which approaches infinity as x approaches infinity. For all real a > 0, r > 1 and j, t E N there exists a standard natural number m such that for all n > m , whenever I is an interval of length n and any nonempty set A C I has the m, r density property on I then A contains a u, (3 uniform < t, a, d, w > cell with {1; < h('!f), '![ < h( � ), (3 < h( � ) and g(::r,) < d < y · Furthermore, we may take w and d to be powers of 2. Proof. ( Sketch only ) . Suppose h (x ) , g(x ) , a, j, r, t arc given as in the statement above and that no such m exists. By "overspill" there exists an M, N in *N  N with lvl < N and a hypcrfinitc internal set A such that A has the M, r density property on an interval of length N but A contains no < t, a, d. w > cell on this interval with the required properties. Since the conditions are translation invariant we may assume that the interval is [0, N  1] . We now define a standard function f : [0, 1] + [0, 1] by:
f(x) = s t
(
l A n [O, x NJ I I A n [O, NJ I I
)
.
Using the fact that A has the M, r density property on [0, N] it is not diffi cult to show that f(x) satisfies a Lipschitz condition with constant r. Thus, the function f is absolutely continuous, differentiable almost everywhere and equal to the integral of its derivative. Since f(1) = 1, f(O) = 0 and f is the integral of its derivative, it must be that the Lebesgue measure of { x : f' ( x) ::0: ( 1 is nonzero. Thus, there exists a real number c ::0: 1 such that the Lebesgue measure of the set
�)}
E
=
{x : c  �
:S:
j'(x) :S: c
}
1s nonzero. By using the Lebesgue density theorem it is straightforward to show that any set of positive measure contains arbitrarily long arithmetic progressions, and that, in fact, these progressions may have arbitrarily small differences
9. Nonstandard methods and the ErdosTun1n conjecture
136
between elements. This allows us to obtain a < t, a, D, W > cell, with *N  N with the property that there exists B E *N  N such that
st
B D B 2D st st � ) ' � , ) ) ( ( (�
,
D, W E
( � )
. . . , st B + (  1)D
forms an arithmetic progression in E. The a homogeneity follows from the definition of E. The fact that f is differentiable at each point in E allows us to obtain the uniformity condition, and allows us to take U, D and W arbitrarily small but not infinitesimal to N. This, in turn, allows us the freedom to make those quantities powers of 2. We are thus able to obtain a U, uniform < t, a , D, W > cell for A in [0 , N  1 ] with all the properties required in the theorem, contradicting our assumption. D
(3
Definition 9 . 2.4
For A a sequence of positive integers we define the
Banach Density of A or BD(A) by:
ED (A)
=
.
1n f. n1ax x EN{0} a EN
upper
I A n [a + 1, a + x J l . X
Upper Banach density is often simply called Banach density, and is some times referred to in the literature as strong upper density, with notation d* ( A ) in place of B D (A) . That notation is used in [7] . The theorem allows us to obtain some results about the existence of uniform < t, a , d. w > cells in sequences that are relatively sparse (certainly too sparse to necessarily contain actual arithmetic progressions) . The theorem below, also proved in [8] , is of this type.
Let a > 0 and t > 2 E N be given, h ( x ) be any continuous real valued function such that h ( x) > 0 whenever x > 0 and let A be a sequence in N with the property that for all c > 0, l A n [0 , n  1 ] 1 > n l  c: for sufficiently large n. Then for sufficiently large n, A contains a u, uniform < t, a , d, w > cell on [0, n  1 ] with w and d powers of 2 and such that: log d u w w log d 0 and a sequence A satisfying
n
I A n [ 0, n  1J I > e cvlog for sufficiently large n n that contains no 3term arithmetic progression. This result is due to Behrend. For convenience we will adjust the constant and use log base 2 here, and also replace e with 2. By adjusting the constant if necessary (and using 2A) we may assume that A contains no two consecutive numbers. We may also translate so that 0 E A without changing the density condition. Thus, we begin with a sequence A which contains 0 and no two consecutive numbers and satisfies
n
. I A n [0, n  1J I > c .JjOgi1: for sufficiently large n. 2 Let N E * N  N and let (3 =
( 1/ 2 ) 2 1°; mk+ l
mo = N; =
m 1 = the largest power of 2 less than (J( l + o/ 2 ) N
the smallest power of
L = the
2 greater than
smallest number such that
(rr;; )
m£ + 1
::;
0
mk
1.
We now wish t o show by induction that
( 9.2. 1) To see this we note that for k the construction we have
=
1 the definition of m 1 guarantees this. By
9. Nonstandard methods and the ErdosTun1n conjecture
138
so that, assuming the induction hypothesis,
(rr;; ) a mk a 2 (r( l +a/ 2 )k ) p( l +a/ 2 )k N 2 (ra( l +a/ 2 )k ) p( l +a/2)k N 2 (ra/ 2 ( 1 +a/ 2 )k ) (ra/ 2 ( 1 +a/2)k ) p( l +a/ 2 )k N
mk +l < 2
cLylog 2 N
by essentially using block copies of initial segments of *A. More specifically we let and, for
i
1
:::; k
cell on [0, N  1] with '![ < (� t where both w and d are powers of 2 . We show that this forces an actual arith metic progression of length 3 in *A and thus in A (by transfer) , contradicting our assumption about A.
9.2. Near arithmetic progressions
139
To see this, we let i be such that mi+ l :S: d < mi . Then since *A contains no two consecutive numbers, the < 3, d, w > cell on [0, N  1] must be completely contained inside one of the blocks of length mi, i.e. inside some [vmi, (v+ 1 ) mi] · But a
w to is a rejection criterium: to is a constant depending on 8o and a) . For e E 8o, Eerp is the risk of the first kind (at B); for e E 8 1 , Ee (l  rp) is the risk of the second kind (at B); for e E 8, Eerp is the power of rp (at B) . For testing a given null hypothesis, there arc generally a lot of levela tests (for example the constant test rp := a) . A levela test ¢ is said uniformly the most powerful (U.M.P.) if for any levela test rp, ¢ is uniformly more powerful than '1/J , i.e. VB E 8 1 , Ee¢ � Eerp (in fact "more powerful" means "at least as powerful") . U .M .P. tests only exist in particular cases: for example for ! dimensional exponential families if Ho : e :::_: Bo (where Bo E 8, an interval of �) . So, some more sophisticated notions are used to compare the power of two tests: for exemple the relative efficiency ( cf. § 10.5) . It is not possible to summarize this notion in some words; so we suggest to refer to [1] . [2] or [12] .
10.3 10.3.1
Exponential families Basic concepts
The following classical results and more information about exponential fam ilies can be found in [10] or [13] . Let k be a standard integer. We denote by (x I y) = 2::]= 1 Xj YJ the scalar product of x and y, vectors of �k . Let p, be a probability mesure on Sl : = �k (the O"field is the field of borelian sets) and let e
The set
:=
{
eE
�k :
J exp (e I x)p,(dx) < 00} .
8 is convex and for e E 8, let '1/J ( e) = ln
j exp( e 1 x)p,( dx ) .
The function 'ljJ is convex and continuous on 8 ° (the interior of 8). The statistical family { Pe : e E 8} defined by Pe : = PetL where
Pe (x) = exp ( (e I x)  ?jJ(B))
10. Nonstandard likelihood ratio test in exponential families
148
is the (full) exponential family associated to p,. A lot of classical statisti cal families (e.g. multinomial distributions, multidimensional normal distri butions, . . . ) are exponential families, generally after reparametrization. This reparametrization can be chosen such that e0 is nonempty. Let e' := {e E e : Ee i iX II < oo} where 11 1 1 denotes the Euclidean norm. For e E e' we define .>.. ( e) : = EeX ; this mapping is 11 from e' onto A := .>.. ( e') : e' contains e0 and .A is a 11 diffeomorphism from e0 onto A0 (cf. [10, pp.74,75] ) . To see this, notice that for e E e 0 , EeX = \l 'ljJ (e) and all derivatives of 'ljJ exist at e. For e 1 , e2 E e0, e 1 /= e2 , 'lj J is strictly convex on the line joining e l and e2 and then ( e l  e 2 I .>.. ( e l )  .>.. ( e2 ) ) > 0. 10.3.2
KullbackLeibler information number
For eo E e and e E e', the KullbackLciblcr information number is given by 'ljJ (eo)  'ljJ (e) + ( e  eo I .>.. ( e) ) . If eo is a proper subset of e ' let I(e, eo)
=
J(e, eo) := inf {J(e, eo) : eo E eo} . For (�, �o) E A2 , we set and for A C A and � E A, let J ( �, A) : = inf {J ( �, a) : a E A} , J(A, �) := inf {J(a, �) : a E A} . The classical likelihood ratio test of He0 : e E eo against e E e \ eo is based on the statistic
where ( Xi h < i < n is a nsampling of X (sec § 10.2.1). Here. denoting X n
� L Xi i= l
Re0
sup { (e I X)  'ljJ (e) }  sup { (eo I X)  'ljJ (eo) } ()o E 8o () E 8 sup (e  eo I X)  'ljJ (e) + 'ljJ (eo) } . eoinf E 8o () E 8 {
=
10.3. Exponential families
149
If X E A, then >,  1 (X) is the maximum likelihood estimator of e and so
where Ao := >.(eo). A (the closure of A) is the closure of the convex hull of the support of p, (cf. [10] ) ; so, in any case, X E A. As sup {(e I �)  '1/J(B) : e E e} is lower semi continuous with respect to � ' we define (as in [10] ) , for � E 8A (the boundary of A) and �o E A, J(�, �o) by J(�, �o) : = lim inf J( ( , .;o) . e_,�, e EA
So, if X tf. A (then X E 8A) it remains possible to write
Re0 = J(X, Ao). In the following, we shall suppose that, for each �o E A, J(·, �o) is continu ous on A. This assumption is verified by classical exponential families, but it is possible to build counterexamples (cf. exercise 7.5.6 in [10] ) . Note that if �o E A\ hal(8A) is limited, this assumption implies the Scontinuity of J(·, �o) on any limited subset of A. Indeed, if �o is standard, it is obvious. If �o is non standard, and if � is limited, let Bo := >,  1 (.;o); as >. is a diffeomorphism from e 0 onto A0, then oeo = >,  1 (0�o) and so, as 'ljJ is continuous on e0, we can write The Scontinuity of J(·, �o) is then deduced from the Scontinuity of J(·, 0�o). In the following. if A is a subset of � k , A0 will denote its interior, A its closure, aA its boundary for the euclidian topology of �k ; its shadow 0A is a closed standard set (cf. [11, p. 63] ) . If A is a subset of A, §A will denote its boundary for the induced topology of A . 10.3.3
The nonstandard test
In this paper n will be supposed unlimited , so we shall use classical asymptotic properties of the likelihood ratio test ( cf. §10.4) . Let S o be a standard subset of e, such that hal( S o)  its halo  is included in a standard compact K included in e0. Let F : = { eo internal : S o C eo C hal( S o) } and let e 0 be the likelihood ratio test of He0 : e E eo against e tf. eo of size a. Recall that e 0 is defined in the following way: there exists a number d d( a , eo) such that :=
sup Pl) (Re 0
liE8o
> d) :::;
a
:::;
sup Pl) (Re0
liE8o
? d) .
10. Nonstandard likelihood ratio test in exponential families
150
Consequently, if Re0 < d then e0 randomizes if Re 0 = d.
0, if Re 0
>
d then e 0
1 and one
The nonstandard likelihood ratio test (NSLRT) of the non standard null hypothesis (Ho) : e E hal(Go) of level a is defined by
  1 (b) I I < n 1 18 .
D
For example, a standard set A such that A c A0 (e.g. an open standard set) is nregular in any boundary point. It is possible to prove that a limited convex set A such that 0A has a nonempty interior is also nregular in any boundary point (we shall not use this result in the following) .
10. Nonstandard likelihood ratio test in exponential families
152 Lemma 10.3.1
(i) Let (�o , 6) E A2 and let � E ]�o , 6 [ (the segment between �o and 6). Then J ( ( 6 ) < J ( �o , 6 ) . (ii) Let A be a nonempty subset of A and 6 E A0 \ A. Then J(A, 6) J(A, 6 ) = J (aA, 6 ) . (iii) Let E and F be subsets of A such that E is a compact set included in F 0 ; let � E A0 \ E. Then J(F, �) < J(E, 0 and J(�, F) < J (�, E) . =
Proof. (i) As A is convex, � E A. We set �o  � =: a(�  6 ) where a > 0; denoting e = >,  1 (�) and e 1 := >, 1 (6 ) , we have ( using corollary 2.5 in [10]) (ii) Let �o E A. If �o tf. aA, let � E aA n ].;o , 6 [. Using (i), we can write J(�, 6 ) :::; J(�o, 6 ) . So J(aA, 6 ) :::; J(A, 6 ) . Conversely, using the continuity of J ( , 6), we have ·
J (aA, 6 )
�
J (A, 6 ) = J(A, 6 ) .
(iii) If � E F, then J ( P �) = J (�, F) = 0 and J (E, �) /= 0 , J(�, E) /= 0 because � t/. E. We suppose now that � tf. F. Let �E E §E be such that J(E, �) = J (�E , �) ; there exists at least one �F E aF n ]�E , �[ and then, using (i), J (�F, �) < J(�E, �) and so J(F, �) < J(E, �) . Let now �k E §E be such that J (�, E) J (�, �k) ; there exists at least one �� E §F n J �k , �[. We set B E := A  1 (�k) and Bp := >.  1 (�� ). Then =
J (�, �k)  J (�, �� ) = =
1/J (BE)  1/J (B) + (e  e E 1 �)  1/J (BF) + 1/J (B)  (e  eF 1 �) = '1/J (e E )  1/J (eF) + ( eF  e E 1 �) = 1/J (BE)  1/J (BF) + (eF  e E 1 �� ) + (eF  eE 1 �  �� ) = 1 ( eF , e E) + ( eF  e E 1 .;  �� ) .
Then setting .;  �� = : (3( ��  �k) where (3 > 0, we can write according to corollary 2.5 in [10] .
D
10.3.
Exponential families
153
Proposition 10.3.2 Let A be a limited subset of A \ hal (8A) , 6 = >. (B1) be a limited point of A \ hal (8A) and �o E aA such that J(�o , 6 ) :: J(A, 6 ) . If A is nregular in �o , then
1

 ln Pl) (X E A) :: J(A, 6 ) , 1
n and if furthermore d(6 , A) 7: 0, then
1

 ln Pl) (X E A) 1
n
=
J (A, 6) ( 1 + 0)
=
@.
Proof. For all (el , e2) E 8 2 ' denoting X (xl , . . . ' Xn ) and X (where Xj = (xj ,i hs i s k E � k ) we can write =
(
=
� 'L.j= l Xj
)
PlJ; ( dx) = exp n ( ( el  e2 I x)  1/J ( el) + 1/J ( e2)) p� ( dx) .
Let eo := >.  1 ( �o) and let 6 =: >. (e2 ) and p > Jn be such that 6 E hal ( �o ) and B( 6 , p) c A. As >. is a standard diffeomorphism from 8° onto A0 , Bo , B1 , B2 are limited. Then
PlJ; ( X E A) > PlJ; (X E B ( 6 , p)) > c exp n ((el  e2 I X)  1/J(Bl ) + 1/J(B2)) dP� jXEB(b,p) > exp n c ((el  e2 I X)  '1/J (Bl ) + 1/J (B2)) dPlJ2 JXEB(b,p)
(
(
)
)
by Jensen ' s inequality. P� (Xi E 6 ,  p/ 2, 6,; + p/2]) = + 0 for According to (8.4) in each i = . . . k. Then the nonstandard law of large numbers yields P� (X E B ( 6 , p)) = + 0 and we can write ( denoting by 1:k the external set of limited vectors of �k )
[ 4] ,
1
1 1  ln P(j ( X E A) n
1
>
[
1
,
.!. ln( 1 + 0 )
(B 1  e2 1 6 + J:k p )  1/J(BI ) + 1jJ(B2) + n 0 ( e l  e2 1 6) + J:p  1/J(e 1 ) + 1/J(e2) + n  1 (e2 , e1 ) + 0.
As
I (B2 , B1 )  I(eo , B 1 ) = (eo  e1 1 6  �o ) + I (B2 , Bo), and I (B2 , Bo) = @ I I B2  Bo ll = @ 11 6  �o il with I I Bo  B 1 ll limited, 11 6  �o il we have ( cf. lemma 3. 2 . 2 in
[ 13] ),
::
0
 I (B2 , e1 ) =  I(eo , el ) + 0 =  J(A, 6) + 0 ,
10. Nonstandard likelihood ratio test in exponential families
154
and so

1 n
 ln P� (X E A)
2'
 J (A, 6 ) + 0.
Conversely, the limited set A is included in a hypercube [ p, p] k where p is a standard integer; pave it with n k (2p) k hypercubes ( T1 ) with side 6 = � · Among these hypercubes, eliminate the ones which do not intersect A and for the others choose t1 E A n Tt . In the aim of simplicity, we shall denote again the selected hypercubes by ( Tt h< t < N · Let Bt : = >. 1 ( tt) , Bt is limited because t1 is limited and t 1 tf. hal(8A) . F�om the relation N < nk (2p) k we deduce In N = In n £ . Then we can write ' n n N dPenl p� ( X E A ) :::: L r l=l Jp<E Tl } N L r X E T1 } exp n( ( el  el I X)  1/J (Bl ) + 1/J (Bt)) dP� l=l }{
(
<
Jn be such that B ( 6 , P) C A and p IIB2 II :: 0. As in the proof of proposition 10.3.2, we can write 1 n ln Pl)1 (X E A) > (B 1  e2 1 6 + £kp)  1jJ (B 1 ) + 'ljJ (B 2 ) + _!_n ln (1 + 0 ) (e l  e2 1 6 ) + £p + 0  1/J (B 1 ) + 1/J (B2 ) + 0 .J( 6 , 6 ) + 0 = J( �o , 6) + 0 =  J(A, 6 ) + 0 because 116  �o il :: 0 and .J( · , 6 ) is Scontinuous. Conversely, as in the proof of proposition 10.3.2 , use a paving (T1h < l < N with side 6 = � where each T1 is closed and such that T1 n A /= 0 , but cho;se t1 in T1 (maybe outside A) such that: if B1 , z  el. z > 0 then t l , z is maximal in T1 if B1 , z  el. i � 0 then tl , i is minimal in T1 So, for X E Tl we can write (e l  el I X) :::.: (e l  el I tl) and then
Pe� ( X E A)
N
�
L Jrp<E Tz } dr'e 1
.(8o). We shall see in lemma 10.3.2 that J(· , Ao) is continuous; then if c < c' < co , Ac is a proper subset of Ac' (for if not, { � E A : J(�, Ao) :::; c1c' } = { � E A : J(�, Ao) < c1c' } is a connected component of A which is a connected set.) Lemma 10.3.2
If c :::; co is limited, then aAc is a limited compact set.
= { � E A : J ( �, Ao ) = c} is closed in A since the function � J(�, Ao) is continuous on A as we prove now. By transfer, we just have to prove this continuity for a standard Ao: if 6 and 6 are such that 11 6  6 11 ::: 0, if �o E Ao is such that J(6 , �o) ::: J(6 , Ao) then, as J(, �o) is Scontinuous, J( 6 , �o) ::: J(6 , �o) and so J( 6 , Ao);;,J(6 , Ao) ; similarly J(6 , Ao);;,J( 6 , Ao). We prove now that if � is unlimited, then J(�, A0 ) is unlimited. For each �o E Ao , �o belongs to the standard compact K; then II �  �o il ::: oo and so J(�, �o) ::: oo (cf. [10, p. 177]). Thus J(�, Ao) is unlimited. Therefore { � E A : J(�, Ao) = c} is limited. D ____,
Proof. a Ac
Proposition 10.3.4 Let c :::; co be limited and B1 be a limited point of 8' such that 6 := >.(B 1 ) tf. hal (8A) . (i) If 6 tf. Ac, then � ln PJ: (X E Ac) = � ln PJ: (X E A�) :::  J( aAc, 6 ) .
(ii) If 6 E Ac, then � ln PJ: ( X E Ac)
:::
(iii) If 6 tf. Af , then � ln PJ: ( X E Af)
0.
=
� ln PJ: (X E Af) ::: J( aA c , 6 ) .
(iv) If 6 E Af , then � ln PJ: ( X E Af) ::: 0.
10.3. Exponential families
157
Proof. We first prove the both similar results (i) and (iii), and then the both similar results (ii) and (iv) which are obtained in a same way. (i) If 8o and c are standard, Ac is a standard compact set such that Ac A2 . Indeed, A� {� E A : J (�, Ao) < c}. If � is such that J (�, Ao) c , let >.. o E Ao (a compact set) be such that J(�, Ao) = J (�, Ao) = J (�, >..o ) = c. According to lemma 10.3.1 (i) , for any ( E ] ( >.. o [, J ((, >..o ) < J(�, >..o ) and so J((. Ao) < c. Thus ]�, >..o [ E A� and then � E A2. So, according to proposition 10.3. 1 , Ac and A� arc nregular in any point of 8Ac and then, ac cording to proposition 10.3.3, we can write � ln PlJ; (X E Ac) :::::  J (Ac, 6 ) and � ln PlJ; (X E A�) ::::: J(A� , 6 ) . Finally, lemma 10.3.1 (ii) yields J(Ac, 6 ) = J (A�. 6 ) J(aAc. 6 ) . Now, if c or 8o are not standard, the continuity of J (�, ) on >.. ( K) implies 0Ac(8o) Aoc(08o) and the hypothesis of proposition 10.3.1 is verified since =
=
=
=
·
=
hal (8Ac)
=
hal (aAc) U { � E 8A : J (�, Ao) � c}
and
hal( a 0Ac) = hal (a0Ac) u { � E aA : J (�, 0Ao) � °C} . Indeed, on one hand, the continuity of J (�, ) implies ·
and on the other hand, this continuity also implies hal( 8Ac)
{ � E A : J (�, >.. ( 8o)) ::::: c} { � E A : J(�, >.. ( 08o)) ::::: 0c}
=
hal (8 °Ac)
·
Then we can conclude as before, using proposition 10.3.1 , proposition 10.3.3 and lemma 10.3. 1. (iii) In the same way, Af is nregular in any point of aAf = aAc, but Af is not always limited. Meanwhile, aAc is limited and so the proof (in proposition 10.3.3) of � ln PlJ; (X E Af ) ?  J (8Ac, 6 ) + 0 remains valid. Suppose now that 6 E Ac. Ac is included in a standard hypercube Up : = {x E �k : Xi > p} [  p , p] k (where p is a standard integer) . If we set Sp,i and s�, 2 {X E �k : Xi <  p } then u:; u7= 1 (Sp,i u s�, i ) . I t i s clear that :=
=
:=
On one hand, as Up \ A c is limited and nregular in any boundary point, we can write
10. Nonstandard likelihood ratio test in exponential families
158
Using lemma 10.3.1 (iii) with E = §uP and F = A( , we have J(aUPu aAc, 6 ) = � � 6). J(8Ac, 6) and so n1 ln Pe 1 ( X E Up \ Ac) ::::: J(8Ac, O n the other hand, n
P8; ( X E Upc ) �
k
2._) P8; ( X E Sp,i) + P8; ( X E S�,i )) . i =1
We know that � ln P8; (X E Sp ,i) ::::: J(Sp,2 , 6 ) and � ln P8; (X E 3� ,2) :::::  J(S�,i ' 6) (it is a classical result concerning halfspaces: cf. chap. 7 in [10]); so 1
1  E U C ) � max max(  J( SP 6 ) , J( S , 6)) + 0  ln P8; (X ·"' P p,; n 2=l... k which is less than  J(aAc, 6 ) according to lemma 10.3. 1(i). Then � ln P8; (X E � J(8Ac, 6 ) + 0. 1 C � · Fmally, r;, ln Pe 1 ( X E Ac ) ::::: J(8A c, 6 ). (ii) If 6 E hal(aAc) , then exists �o E §Ac such that J (�o , 6 ) ::::: J(A, 6 ) ::::: 0 and it remains possible to use proposition 10.3. 1 , proposition 10.3.3 and lemma 10.3.1 as for the proof of (i). If 6 tf. hal(aAc) , then 6 E 0Ac \ hal(aAc) according to the proof of (i) and so there is a standard number p such that the ball B (�, p) is included in 0Ac \ hal(§Ac) and consequently included in Ac. Then the nonstandard law of large numbers yields P8; ( X E Ac) = 1 + 0 and finally ln P8; ( X E Ac) ::::: 0. (iv) is obtained in a similar way. D A
A cc )
n
Definition 10.3.3 Let e E 8'. Let he be the function defined on [0, co [ by h e( c ) := J(A(, >.(B)) and ge the function defined on [0, co [ by ge ( c) : = J(Ac, >.(B) ) . ___,
As c Ac is increasing, ge is a nonincreasing function and he is a non decreasing function. Let r(B) : = I(e, 8 0 ); >. ( e ) E Ac iff c ? r(B) and so ge ( c ) = 0 if and only if c E [r(B) , co [ and h e( c) = 0 if and only if c E ]o, ,(B)].
Proposition
10.3.5 ___,
___,
h e (c) are Scontinuous on (i) The functions (e, c) ge (c) and ( e, c) {e E 8 \ hal(88) : e limited } x ( [O , co [ n £). (ii)
The function ge is decreasing on [0 , /( B)] and he is increasing on [r(B) , co] .
10.4. The nonstandard likelihood ratio test
159
Proof. Let e be standard and e ' ::: e , then .>.. ( e ) ::: .A ( e ' ) ; let c and c' be such that c' ::: c. Then °Ac' = 0Ac and Similarly, J(Ac' ' .>.. ( e' )) ::: J(0Ac' ' .>.. ( e)). The monotonicity of ge is deduced from lemma 10.3.1 (iii) which shows that if c' < c and .A ( e ) tf. Ac' then J(Ac' , .>.. ( e)) > J(Ac, .>.. ( e)). D The continuity and monotonicity of he are proved in the same way. 10.4
The nonstandard likelihood ratio test
Recall that F := { 8o internal : So c 8o c hal(So) } , that the NSLRT is defined by d) :S: a :S: eEe eEe o o (see §10.2.1) can be written : ==:
sup Pl) ( X E Ar (8o)) eEe o
:S:
a :S: sup Pl) ( X E Ar (8o)). eEeo
Consequently, Ar is the rejection set (relatively to X ) , A� the acceptation set and the test e 0 is randomized if X E aAc . As n is illimited, if d < co (8o) and if d is limited then, for e E 8o, proposition 10.3.4 yields
As supeE e o J(aAd, .>.. ( e ))
=
d by definition of Ad, we have d(a, 8o) ::: ln a .  
n
This is the nonstandard form of a classical result of Bahadur ( [1] , [2] ) . We shall study the NSLRT according as these d( a, 8o) are infinitesimal or appre ciable. In this latter case, we shall have to distinguish the cases d( a, 8o) < co (8o) and d(a, 8o) ? co (8o). In the following, Zo stands for co (So), Ao for ).. (So) and Ro J (X, Ao) for R e 0 ; Ac will denote Ac ( Ao); the notations he (c) and ge (c) will also refer =
10. Nonstandard likelihood ratio test in exponential families
160
to these Ac implies
:=
K
Ac (Ao). Note that if
8o E
F, then the Scontinuity of A on
�
Re0 Ro. The following lemmas will be used for the calculation of the risks of the first and the second kind. They are valid not only for exponential families, but also for any statistical family. Here 8" is a standard subset of 8. Lemma 10.4.1 Let T be a statistic, S a (perhaps external) subset of 811 and F : D ____, � + , ( e , c) ____, Fe (c) a standard continuous function defined on a domain D = {(B, c) E 8" x �+ : de � c � d� } such that e • •
the functions e + de and e + d� aTe continuous, for each e E S, Fe is decreasing on [ de , d�] , for each e in s and c nearly standard in l de ' d� [ ' 1 n ln Pe(T � c) � Fe ( c) .
Then, for each e nearly standard in S and Co standard in [de, d� [ Proof. PRIJ (T � Co) = supw =@ Pe (T > Co + w). Now write 1  ln Pe (T � Co + w) � Fe (Co + w) � o Fe (Co + w). n
So, as oFe = Fe0 ( where Bo we have
:=
oe) is a standard continuous decreasing function,
� ln PR(j (T� C0 ) = J oo, ��� Fe (Co + w) [ = ] oo, Fe (Co) + 0[ .
Similarly, 1 n ln PR e (T � C0 ) = ] oo, ° Fe (Co) + 0[ = ] oo, Fe (Co) + 0 [ , n
for if not, there exists T) < 0,
T)
�
0 such that
1
Vr:: E hal + ( o ) , n ln Pe(T � Co + r:: ) > o Fe (Co) + TJ
where hal + ( o )
:=
{c: E hal ( O ) : � 0} . r::
10.4. The nonstandard likelihood ratio test
161
Then { E � + : � ln PlJ (T ? Co + ) > ° Fe (Co) + 7]} is an internal set in cluding hal + ( o ) and consequently ( by Cauchy ' s principle ) including an interval [0, wo] where wo is standard. But c:
c
� ln Pe (T ? Co + wo) :: Fe (Co + wo) :: ° Fe (Co + wo) n with °Fe(Co + wo) < °Fe (Co) which is a contradiction, because both these numbers are standard. Finally D Lemma 10.4.2 Let T be a statistic, S a (perhaps external) subset of 811 and G : E + � + , ( e , c) + Ge (c) a standard continuous function defined on a domain E = { (B , c) E 8" x � + : e e ::; c ::; e� } such that •
the functions e
____,
ee
and e
____,
e� are continuous,
•
for each e E S, Ge is increasing on [ee , e�] ,
•
for each e in s and c nearly standard in l ee ' e� [ ' 1  ln PlJ (T ::; c) :: Ge (c) . n
Then, for each e nearly standard in S and Co standard in [e e , e � [ PRIJ (T � Co) = L: e nGe ( Co ) + n0 . Proof. PR� (T � Co)
=
inf t =©: PlJ (T ::; Co + t). Now write
� ln PlJ (T ::; Co + t) :: Ge (Co + t) :: 0Ge(Co + t). n So, as 0Ge is a standard continuous increasing function, we have
1 n ln n PRe (T � Co) =
]
 oo ,
]
. Ge (Co + t) = ]  oo , Ge (Co) + ] . 0 tmf =@
As in lemma 10.4.1 , one establishes that PR� (T � Co) finally PRIJ (T � Co) = L:enGe ( Co ) + n0 .
=
PRIJ (T � Co) and D
10. Nonstandard likelihood ratio test in exponential families
162
10.4.1
In a
n
Proposition (i) if Ro
�
infinitesimal
10.4.1
For l�a
0 then
•EIGSI, 17041 La Rochelle, France. scherazade .benhabib@eigs i . fr
11.1. Introduction
171
Then we can define externally the domain of definition of A and its multi plier algebra as follows. Definition 1 1 . 1 . 1
A is the external set
The domain D of definition of the infinitesimal generator
D = { f E � X / f limited and Af limited}
(11.1.1)
And its multiplier algebra is the external set M = { f E D/ Vg E D fg E D}
(11. 1.2)
The topology on X arises naturally from the probability theory and a proximity relation is defined on X, more particulary from the way the functions in the multiplier algebra M of the domain D of the infinitesimal generator act on X. Definition 1 1 . 1 .2
and only if
Let u and v be in X, we say that u is close to v (u Vh E M h(u)
�
�
v) if
h(v)
Thus, the elements of M have a macroscopic property of continuity for mally analogous to the Scontinuity. With this relation, we define the external sets :
�
Cx
=
X}
(11. 1.3)
Fx
= {y E X y rf. X}
( 11.1.4)
{y E X y
of points close to x and :
of those far from x. For each x in X, (a(x, y)) are positive real numbers if x /= y and can take unlimited values. Let Ix be the set of all h in M vanishing at x, and for h in Ix let us define ( 11.1.5 ) Ah(x) = L a(x, y) h (y) yE X \{x}
To describe the structure of the semigroup generator, we want to split Ah(x) so that the contribution of the points far from x in X. appears separately. One is tempted to define directly the quantity
aa h (x) = L a(x, y)h (y) yEFx
( 11.1.6 )
11. The infinitesimal generator of a markovian semigroup
172
which alas has no meaning because of the external character of the set of indices F:r Nevertheless, we will show in Section 1 1.2 how we can attach a definite meaning to it. That leads in Section 1 1 .3 to a definition for an axintegrable function. Theorem 11.3.1 establishes the representation of Ah(x) for functions in M which have a zero at x of the third order. This is the global or integral part of A. Theorem 11.3.2 and its corollary extend the notion of axintegrable to functions which have a zero of the first or the second order. 11.2
Construction of the least upper bound of sums in 1ST
Let E be an external subset of X and f a positive function defined on X. If there exists a standard natural number no such that for all internal sets W contained in E we have
Lemma 1 1 . 2 . 1
L f(y) :S: no
(11.2.1)
yEW
then there exists a unique up to an infinitesimal least number a( ! ) such that
L f(y) � a( ! )
(11.2.2)
yEW
for all internal sets W contained in E. Proof. Let no be a standard natural number such that for all internal sets W contained in E we have ( 11.2.1). By external induction, there is a least no such that (1 1.2.1) holds. Now for each standard natural k there is a largest natural number j with 0 :S: j :S: 2 k such that for all such W (11.2.3) L f(y) :S: no  k yEW
1
Let a(k) no  ik . Then for each standard k we have a standard a ( k ) , so there is a standard sequence k + a( k) . This sequence is decreasing, therefore it has a standard limit a (f) . Moreover, for all standard k and all such W , D ( a (f)  Ly Ew f(y)) :S: 21k · This proof uses very elementary tools. It can easily be formulated in the minimal nonstandard analysis introduced by Nelson in [3] in which "standard" =
1 1.2. Construction of the least upper bound of sums in 1ST
1 73
a(O)
Step 0:
no  1 Step 1:
no  1
/,� i)\ : : J)\ n
(2
Step 2:
no  1
no a(1) no a(2) no
Figure 11.1 applies only to natural numbers. On the contrary, if all the force of 1ST is used, one could check that s a (f) sup x E �+ ; 3 intw C E L f(y) :S x
=
{
yEW
}·
Figure 11.1 exhibits the way the sequence is built. Definition 1 1 . 2 . 1 1.
If f is a positive function defined on X, and if the conditions required for Lemma 1 1 . 2. 1 are verified the quantity (11.2.4) af = L f(y) yEE
is defined to be the standard limit a (f) . 2. If the conditions for Lemma 1 1 . 2. 1 are not verified, let af be the formal symbol x and say that (1 1 . 2. 2) is not defined. 3. In the general case, let f f +  f  be the decomposition of f into the difference of two positive functions. Say that af is defined in case both aJ + and a1 are defined, and let a f be their difference:
=
Notation 1 1 . 2 . 1
of af .
For each x in X, if f depends on x, write aJ (x) instead
11. The infinitesimal generator of a markovian semigroup
174
11.3
The global part of the infinitesimal generator
For x E X and h E M , let f = a(x, ·)h, then h is said to be axintegrable if and only if aJ (x) is defined.
Definition 1 1 . 3 . 1
Notation 1 1 . 3 . 1
In that case write aa h (x) instead of aJ (x) .
Definition 1 1 .3.2 1.
h is a function in Ix if and only if h E M and h(x)
2. h is a function in I� if and only if h E M and h limited and eb gk ·in Ix .
= 0.
=
I:f=l e k gk with p
3. h is a function in I� if and only if h E M and is of the form h Ll e k fk gk with e k , fk , gk E Ix and p limited. 4. We say that h has a zero at x of the first order when h E Ix, of the second order when h E I�, of the third order when h E I� . If h has a zero at x of the third order, then h is axintegrable and Ah ( x) is infinitely close to aah ( x) .
Theorem 1 1 . 3 . 1
Proof.
The proof goes through the following three steps:
1 . Let h be in I� , therefore it is of the form h and fk , gk in Ix .
=
I:i f� gk with q limited
,
2. I f f g E I.:r then P g is axintegrable and so is h. These steps are proved as follows:
1. Let h E I�, as 1 1 1 efg = 4 (e + f) 2 g  2 e 2 g  2 f 2g, therefore h is of the mentioned form.
I L a(x, y)f2 (y)g+ (y) I w
:::;
l l g i i A(f2 ) (x)
1 1.3. The global part of the infinitesimal generator
175
As l l g l l and A(f 2 ) are limited, they are both infinitely close to a standard number. Taking the integer part of o (l lgi iA(f2 ) (x)) plus 1, we get the existence of the natural number no required by Lemma 11.2.1. The same argument applies to f 2g  . Thus aa( J2 g + ) (x ) and aa(j 2 g) (x ) exist and as O!a( J 2 g) (x ) = aa( J 2 g + ) (x )  aa(j 2g) (x ) , then f2 g is ax integrable, and so is h. 3. Let (3 » 0. Since g E Ix there is an internal set W of Fx such that g + (y) :::; (3 for y E W 0 . We have
I A(f2 g + ) (x)  aa( J 2g+) I
=
IL
yE X \{x}
a (x , y) f2 (y)g + (y)  aa( J 2g + )
I
L a (x , y)f2 (y)g+ (y)  L a (x , y)f2 (y)g+ (y) L L a(x, y)f2 (y) :::; (3 L a (x , y) f 2 (y)
0 and c, N > 0 . Consider an adapted probability space (r, Q, IP') and B[O, 1 ] ® Y1measurable geometric Brownian mo tions with constant multiplicative drift g� : r x [0, 1 ] �d·n , i E { 1 , . . . , n}. Then there exist processes gi , i E { 1, . . . , n} on an adapted probability space (D, F, CQ) , equ.ivalent to the processes g� on r (in the sense of adapted equiv alence [7}), such that there is a probability measure Mm on n in the class of measure s Q probability measure, VA E F1 iJ · CQ(A) :S: Q(A) :S: N CQ(A) , > vz .../.r. J. E { 1 , . . . , n } 0 I( ) I s c minimising ____,
1 1 CovQ((g,)s, (gJ)Jd .t.Q 9t9J s
}
·
w·
_
�
in the class of Loeb extensions of finitely additive measures from (*C) (D, G) (G being an arbitrary lifting of g). Analogously, there is a probability measure Mn minimising
(where nn ( Q, g) is defined to be +oo if the derivative in 0 in this definition does not a.s. exist as a continuous function in t) in the class of Loeb extensions of finitely additive measures from (*C) (D, G) . Moreover, inf, mn (·, g) :S: inf mr ( , g) C(D g )
as well as
C(r,g)
·
inf nr (·, g ) . inf nn (, g) :S: C(r,g)
C(D,g)
12.2. A fair price for a multiply traded asset
1 79
The original paper [8} does not state the Theorem and the subs equent Lemmas as precisely as it is done here, although the exact result becomes clear from the proofs in that paper.
Remark 12.2. 1
The proof for this Theorem can be split into the following Lemmas which might also be interesting in their own right.
Using the notation of the previous Theorem hyperfinite adapted space n [12},
Lemma 12.2.1
12. 2. 1,
for any
inf mn (· , g) :::; inf mr (·. g) . C (r,g)
C(D.g)
Under the asswnptions of Theorem 12.2. 1 and choosing n to be any hyperfinite adapted space, the infimum of mn ( , g) in the class of Loeb extensions of finitely additive measures from (*C) (D, G) is attained by some measure Mm E C(D. g) . Proofs for both of these Lemmas can be found in a recent paper by the author [8] . Although they look very similar, it is technically slightly more demanding to prove the following two Lemmas ( which in turn obviously entail the second half of the Theorem, i.e. the assertions concerned with the map n) . Lemma 12.2.2
Using the notation of the previous Theorem hyperfinite adapted space n [12},
Lemma 12.2.3
12. 2. 1 ,
for any
nr (· , g ) . inf nn (· , g) :::; C inf (r,g)
C(D.g)
Under the assumptions of Theorem 12.2. 1 and choosing n to be any hyperfinite adapted space, the infimum of nn (·, g) on C(D, g) is attained by some measure Mn E C(D, g) .
Lemma 12.2.4
Easy results are Lemma 12.2.5 A semimartingale X is a ?martingale on n if and only if 1 mn (P, x ) = 0 . The function mn (P, · ) v on the space of measurable processes of n satisfies the triangle inequality and is 1homogeneous. For p = 2, it defines
an inner product on the space [.
:=
{x : n
x
[0, 1]
+
�d
:
x
measurable, rn(P, x )
which becomes a HilbeTt space by this construction.
0 is the logarithmic discount rate (assumed to be constant) , b is the onedimensional Wiener process, (Pu)uEI is a convex combination  i.e. I C (0, + oo) is finite, Vu E I Pu > 0 and L uEI Pu = 1. The parameter v depends very much on the tax rate we assume. If p is the logarithmic tax rate and T the expected time during which one will hold the asset (that is the expected duration of the upward or downward tendency of the stock price) , one can compute v as follows: v = T · p. The equation (12.3.1), is of course first of all only a formal equation that thanks to the boundedness of a can be made rigorous using the theory of stochastic differential equations developed by Hoover and Perkins [11] or Albeverio et al. [2] for instance. We will introduce the following abbreviation: n;,(v)  X{ 1 · 1 2: v } a . 'f/ 
·
.
12.4
,
How to minimize "unfairness"
Intuitively, the map nn (P, · ) (for a fixed probability measure P on a proba bility space D) when applied to a discounted asset price process measures how
12. Applications to the theory of financial markets
182
often ( in terms of time and probability ) and how much it will be the case that one may expect to obtain a multiple ( or a fraction) of one's portfolio simply by selling or buying the stock under consideration. For the following, we will drop the first component of n ( indicating the probability measure ) if no ambiguity can arise; for internal hyperfinite adapted spaces and Loeb hyperfinite adapted spaces, we will assume that the canon ical measure ( the internal uniform counting measure and its Loeb measure, respectively) on the space is referred to. First of all we derive a formula that will make explicitly computing n easier in our specific setting: Lemma 12.4.1 If x ( v ) satisfies (12. 3. 1) for some v > 0 on an adapted prob ability space (r, g , CQ), then the discounted price process exp x� v) : t ::0: 0)
( ( )
is of finite unfairness. More specifically,
( ( ))
nr exp xl"l
[ IE [ 1 ,;,1" (x;"1 fr p;x:;1 i)vo) I] dt [ IE [ I d: l .,�o IE [ xi�¥,] + �' I ] dt I
�

The proof is more or less a formal calculation, provided one is aware of the pathcontinuity of our process and the fact that the filtrations generated by b and x(v ) are identical. For this implies that, given t > 0, the value Pro of.
'1/J ( v )
(
)
( ) ( v)u  ""' xt+ 6 Pi X ( t+ u � ) vo ( w ) i EI v
does not change within sufficiently small times u  almost surely for all those paths w where x�v ) (w)  L i E I p� x/��� ) vo(w) tf. { ±v }, this condition itself being satisfied with probability 1 . Now, using this result and the martingale property of the quotient of the exponential Brownian motion and its exponential bracket, we can deduce that for all t > 0 almost surely:
How to minimize "unfairness"
12.4.
183
Analogously, one may prove the second equation in the Lemma: for, one readily has almost surely
In order to proceed from these pointwise almost sure equations to the assertion of the Theorem, one will apply Lebesgue's Dominated Convergence Theorem, yielding
For finite hyperfinite adapted probability spaces, an elementary proof for the main Theorem of this Section can be contrived: Lemma 12.4.2 For any hyperfinite number H we will let for all v > 0 denote the solution to the hyperfinite inztial value problem
x6v) { x ( v )  x (v )  � } (x(v)  "" . x(v) ) X{
\it E
=
0, 0, . . . , 1 t
t+ J, 
x (v)
a
!

t
+ J . 1ft+ J,
6 Pu
uEI .
( tu) VO
2 1 J 1 (2H ! ) 1/'2 2 H !
I
(v )
xt

(u)
L Pu ·X (t u )V O
uEI
I
?: v
}
.� H!
(1 2.4. 1)
12.
184
Applications to the theory of financial markets
(where 1'1£jH ! : n { ± 1 } H ' + { ± 1 } is for all hyperfinite e v}
which immediately follows from the construction of Anderson's random walk [3] B. = v�Jr. H, and the recursive difference equation defining the process x(v) . However,2 this last equation implies Vk < H! Vv > 0
I
[ (v) I ] (v)
J2 1
IE IE X k}i,1 Fk/ H !  X J'r, + 2 H! 1
= 
H!
a ( 1)X
IE
{
(v )
I
((v)
:.:C: v } sv } )
xk/ lf '  L Pu ·X k m u) V O uEl
+ a(1)x
{
X
��� ,  uLE I Pu X ((vm�  u
VO
12.4. How to minimize "unfairness"
185
} "'"' IP' "'"' { 6 XkjH!  6 Pu · X �, u)vo ? v
Now all that remains to be shown is that (v)
k< H! and
(v)
(
uEI
"'"' IP' { XkjH!  "'"' 6 Pu . X );, u)vo 6 (v)
k< H!
(v)
(
uEI
>
X{ X{ X{
i;1
(v ' )
xf/H'  L Pu · X uEI
(v )
xf/ H '  L Pu X uEI
(v )
·
xt/ H '  L Pu · X uE l
1.
This contradicts equation (12.4.6) . Hence, the estimate tablished for all k < H!, leading to (12.4.2). S imilarly, one can prove
(v )
(t
w u
(v )
(f
wu
(v )
w_  u
(t
)VO ) VO ) VO
(12.4.3)
2:v ' } (w) 2:v' } (w) 2:v } (w)
has been es
(v)  L Pu · X (v ),  O ::; v } L IP' { Xk/H' ( H u)V k�R uEI "" IP' { Xk/H' "" Pu . X((v')l,) u)vo ::; ' } (v' )  L._., ::; L._., k�R uEI which entails that Lk < H! IP' { Xk/�!  'I:, Pu · X (v1 _ u)vo ::;  v } must be monouEI ( H ! Vv' ::; v W
0 ) is a family of stochastic processes on an adapted probability space r such that y(v) solves the stochastic differential equation (12. 3. 1) formulated above for all v > 0. Then the function J f+ nr (y(" l ) attains its minim·um on [0, S] in S.
Theorem 12.4.1
Pro of. B y the previous Lemmas 12.4.2 and the formula for n from Lemma 12.4. 1, the assertion of the Theorem holds true internally for hyperfi nite adapted spaces r, if we replace y(o) by a lifting y(o) and if we let nn when applied to internal processes denote the hyperfinite analogue of the standard nn . Now, according to results by Hoover and Perkins [11] as well as Albeverio
187
References
et al. [2] , the solution x(v ) of the hyperfinite initial value problem (12.4.1) is a lifting for the solution x(v ) of ( 12.3.1) on a hyperfinite adapted space for any v 2: 0. Now, y c+ nn ( y ) is the expectation of a conditional process in the sense of Fajardo and Keisler [7] . Therefore, due to the Adapted Lifting Theorem [7] , we must have Vv 2: 0 °nn x(v ) = nn ( x(v ) )
( )
(where we identify n with its internal analogue when applied to internal pro cesses) . Since the internal equivalent of the Theorem's assertion holds for internal hyperfinite adapted space, the previous equation implies that it is also true for Loeb hyperfinite adapted spaces. Now let (y (v ) : v > 0) be a family of processes on some (not necessarily hy perfinite) adapted probability space r with the properties as in the Theorem . Because o f the universality o f hyperfinite adapted spaces [7, 12] , we will find a process x (v ) on any hyperfinite adapted space n such that x and y are automor phic to each other. This implies [2] that x ( v ) satisfies ( 12.3.1) as well. Further more, as one can easily see using Lebesgue's Dominated Convergence Theorem, Vv
>
0 nn ( x ( v) )
=
nr ( y ( v ) ) .
Due to our previous remarks on the solutions of (12.3.1) on hyperfinite adapted spaces, this suffices to prove the Theorem. D Acknowledgements. The author gratefully acknowledges funding from the German National Academic Foundation ( Studienstiftung des deutschen Volkes ) and a predoctoral research grant of the German Academic Exchange Service ( Doktorandenstipendi1Lm des Dwtschen Akademischen A 1Lstauschdienstes ) . He is also indebted to his advisor, Professor Sergio Albeverio, as well as an anony mous referee for helpful comments on the first version of this paper.
References
[1]
S . ALBEVERIO, Some personal remarks on nonstandard analysis in prob ability theory and mathematical physics, in Prnceedings of the V!Ilth In ternational Congress on Mathematical Physics  Marseille 1986, eds. M . Mebkhout , R. Seneor, World Scientific, Singapore, 1987.
[2]
S . ALBEVERIO , J. FENSTAD , R. H0EGHKROHN and T . LINDSTR0M ,
Nonstandard methods in stochastic analysis and mathematical physics, 1986.
Academic Press, Orlando,
[3]
R. M . ANDERSON, "A nonstandard representation for Brownian motion and Ito integration", Israel .Journal of Mathematics. 25 (1976) 1546.
188
12. Applications to the theory of financial markets
[4] A. CoRCOS ET AL . , "Imitation and contrarian behaviour: hyperbolic bubbles, crashes and chaos", Quantitative Finance, 2 (2002) 264281. [5] N. CUTLAND, E . KoPP and W . WILLINGER, "A nonstandard treatment of options driven by Poisson processes", Stochastics and Stochastics Re ports, 42 ( 1 993) 1 1 5 133. [6] l\1 . D REHMANN, .J . 0ECHSSLER and A . RoiDER, Herding with and with out payoff externalities  an internet experiment , mimoe 2004. [7] S . FAJARDO and H. J. KEISLER, Model theory of stochastic Lecture Notes in Logic 14, A. K Peters, Natick (Mass . ) , 2002.
processes,
[8] F . S . HERZBERG, "The fairest price of an asset in an environment of tem porary arbitrage", International Journal of Pure and Applied Mathemat ics, 18 (2005) 121  131. [9] F . S . HERZBERG, On measures of unfairness and an optimal currency transaction tax, arXiv Preprint math. PR/0410543. http : //www . arxiv . org/pdf/math . PR/04 10543 [10] D . N . HOOVER and H . .J . KEISLER, "Adapted probability distributions", Transactions of the American Mathematical Society, 286 ( 1 984) 1 59201. [ 1 1] D . N . HooVER and E . PERKINS, " Nonstandard construction of the stochastic integral and applications to stochastic differential equations I, II", Transactions of the American Mathematical Society, 275 ( 1 983) 158. [12] H. J. KEISLER, lnfinitesimals in probability theory, in Nonstandard anal ysis and its applications, ed. N. Cutland, London 1Iathematical Society Student Texts 10, Cambridge University Press, Cambridge, 1988. [13] P . A . LOEB , " Conversion from nonstandard to standard measure and ap plications in probablility theory", Transactions of the American Mathe matical Society, 2 1 1 ( 1 974) 1 13122 . [14] E . NELSON, Radically elementary probability theory, Annals of Mathemat ics Studies 1 17. Princeton University Press, Princeton (NJ) , 1987. [15] E. PERKINS, Stochastic processes and nonstandard analysis , in Nonstan dard analysisrecent developments, ed. A.E. Hurd, Lecture Notes in Math ematics 983, SpringerVerlag, Berlin, 1983. [16] K . D . STROYAN and .J . M . B AYOD, Foundations of infinitesimal stochastic analysis, Studies in Logic and the Foundations of Mathematics 1 1 9, North Holland, Amsterdam, 1986.
Quantun1 Bernoulli exp eriments and quantum stochastic processes Manfred
\iVolff*
Based on a
W ...algebraic
Abstract
approach to quantmn probability theory we
construct basic discret e in terna l quantum stochastic pro cesses with in
d ep eiHlent increments. We ob tain Bernoulli
exp eriments
a oneparameter family of (classical)
as linear combinations of these basi c processes .
Then we use the nonstandard hull of the internal GN SHilbert space corresponding to the chosen state
T
( the
?{,.
un derlying quantum probabil
i ty measure) in o rder to derive nonstandard hulls of our internal pro cesses.
Finally continuity requirements lead to the specification of a
certain flUhf;pace £ of
nal pro cesses to
the
can
Hr
to which the nonstandard hulls of our inter
be restricted and
wh ch
i
turns out to be isomorphic
LoebGuicltardet space intro d uced by Lci tzMartini
space of £ the.n ifl shown to he isom orphic to
F+
(L2 ([0, 1] , >.))
1 101.
A sub space
flymmetric Fock
and om· basic processes agree with the processes of
Hudson and Pa.rtltasarathy on this
13. 1
t.he
subspace.
Introduction
In the early 1980's Hudson and Parthasarathy [5], Hudson and Lindsay [4] . and Hudson and Streater [6] started the theory of quantum stochastic pro cesses, in pcu·ticular of quantum Brownian motion. In [12[ Parthao:;arathy showed one way how to make the passage from quantum random walk to diffu sion. All theS(' topics are extensively treated in P.A. Meyer's compendium [ 1 1 [ . For another general introduction to this field see fl3] where an extensive mo tivation ti·om qua.ntum physics may be found. The HudsonPart.hasarathy ap proach is based on t.he classical symmetric Fock space over L2 (R+ , >.) where ·Mathematisches Institut, EberhardKarlsUniv. Tiibingen, manfred . wolff�unituebingen . de
190
13.
Quantum Bernoulli Experiments
A denotes the Lebesgue measure (see section 13.6 ) . Guichardet [3] gave an interesting representation of this Fock space as an L 2 direct sum of spaces L� = L 2 (Xn) where Xn is the space of all subsets of the unit interval [0, 1] having exactly n elements. The measure on Xn is defined via the conesponding Lebesgue measure on �f , This representation was extensively used by Maassen developing his kernel approach to general quantum stochastic processes. Journe [7] invented the so called toy Fock spaces (bebe Fock in French) as discrete approximations of the symmetric Fock space. Let Do = { 0, 1} be equipped with the uniform distribution fLO · Then the toy Fock space of order n is the space L 2 (D0, p, � n) . A rigorous discrete approximation of the Guichardet space and the basic quantum stochastic processes defined on it which uses the N space L2 (D � , p,� ) as a toy Fock space is given by Attal [2] . There is another approach based also on the Guichardet space to prove such approximation theorems using nonstandard analysis which was developed by LeitzMartini [10] . He discretized the Guichardet space in the following manner: let T = { � : k E *Z , 0 :::; k :::; N } be the hyperfinite time line where N E *N is infinitely large. Let r n be the set of all internal subsets of T of internal cardinality n and set r = u�=O rk . Now let ]1.1 be an internal subset of r and set where IAI denotes the internal cardinality of A. Then m is an internal measure on r with m(f) � e. The space L 2 (r, Lm) , where Lm denotes the Loeb measure associated to m, contains the Guichardet space as a Banach sublattice. In fact L 2 (r, Lm) is the nonstandard version of the Guichardct space, so to speak the Loeb Guichardet space over the time interval [0, 1] , Among other things LeitzMartini constructed all the relevant discrete quantum stochastic processes and showed that their nonstandard hulls exist in a precise manner and form the basic quantum stochastic processes: the time process, the creation and annihilation processes, and the number process. All approaches mentioned so far start with some kind of discrete approxi mation of the symmetric Fock space or the isomorphic Guichardet space over �+ , [0 , 1] respectively. In contrast to these approaches we present here a new one based on the more natural W* algebraic foundation of quantum stochastic as described e.g. in [8] , In [13] the author sketched a different though similar approach in the finitedimensional setting but he was not able to make the transition from the finitedimensional to the continuous infinitedimensional case. Our access is exactly the noncommutative version of Anderson's way [1] to Brownian mo tion. In particular we arc able to ju.s tify the usc of the symmetric Fock space (or equivalently the Guichardet space) in the context of quantum stochastic
191
13.2. Abstract quantum probability spaces
processes. We believe that by our approach the applications of nonstandard analysis as given by LeitzMartini [ 10] are also better understandable. We begin with the discrete quantum Bernoulli experiment modelled in the von Neumann algebra of all 2 n x 2 nmatrices over CC. Then we prove a l\loivreLaplace type theorem for quantum Bernoulli experiments, i.e. we show that these discrete stochastic processes converge to quantum Brownian motion in the same sense in which the classical approximation of the usual Brownian motion is treated by Anderson [ 1] . Furthermore we construct the basic quantum stochastic processes out of the corresponding discrete versions in the case of a vacuum state. Finally we show how the symmetric Fock space approach fits into our setting. Applications to the theory of stochastic processes are under preparation .
13 . 2
Abstract quantum probability spaces
Recall that almost all information about a given probability space (D, I:, JL ) is to be found in the pair (A, JL) with A = L= (n, I:, JL). Let us denote by [! ] the equivalence class modulo negligible functions in which the essentially f( w ) dJL( w ) = : bounded measurable function f is contained. Then [f] + JL(f) defines a positive, linear, order continuous normalized functional on A. Analogously a quantum probability space is a pair (A, T) where A denotes a W*algebra and T is a positive, linear order continuous normalized functional on A, or a normal state for short. The reader not familiar with the abstract theory of W*algebras may look at them as subalgebras of the algebra of all bounded operators on an appropriate Hilbert space which arc closed under involution * : a + a* and which are also closed with respect to the strong operator topology. In this context order continuity means the following: whenever a downward directed net (aa) of nonnegative selfadjoint operators from A converges to the operator 0 with respect to the strong operator topology then lima T(aa) = 0 holds. In particular one may interpret L= (n, l::JL ) as the W*algebra of all multiplication operators MJ : g + MJ (g) = fg (! E L= (n, I:, JL)) on the Hilbert space L 2 (D, I:, JL ) . Let (A, T) be a quantum probability space. Let A1 , A2 be W*subalgebras generating the W*subalgebra A3; moreover let Tk be the restriction of T to Ak (k = 1 , 2, 3). Then A1 and A2 are called stochastically independent if (A3 , T3) is isomorphic to (A1 ®A2 , Tl ®T2) where ® denotes the W*tensor product .
fn
Remark: Let us point out that this notion of independence agrees with the usual one in the classical (commutative) case A = L= (n, I:, JL) but that
13. Quantum Bernoulli Experiments
192
there are other notions of independence in the quantum stochastic setting also generalizing the classical one (e.g. free independence) . Two probability spaces (A. T) and (A', T1) are called equivalent if there exists an order continuous algebraic isomorphism rp : A + A' with T1 = T o rp . Now let us construct the L 2 space corresponding to the probability space (A, T). It is nothing else than the so called GNSspace (after Gelfand, Naimark, and Segal) . To this end we introduce the Cvalued mapping (al b) = T(a*b) on A x A. It is scsquilincar, that means, it is linear in the second argument, antilinear in the first argument, and it satisfies (a l a) 2 0 for all a E A. The space LT = {a : (a I a) = 0} is a left ideal, and on A/ LT there is welldefined a scalar product by (a l b) = ( a l b) where a = a + LT denotes the equivalence class in which a is contained. 7{T is the completion of A/ LT with respect to the associated scalar product norm 11 a11 = � = JT(a* a) . The representation JrT : A + J:(HT ) is given by JrT (a)(b + LT) = ab + LT which is welldefined since LT is a left ideal. The representation is injective whenever A is simple (i.e. has no nontrivial closed ideals) , e.g. A = Mn (C), the algebra of all n x nmatrices. In case A = L = (n, p,) the space 7{T turns out indeed to be the space 2 L (D, p,) and in this way L = (n, p,) is represented as the algebra of operators of multiplication by L =functions (see above) . 13 . 3
Quantum Bernoulli experiments
Let us begin with the easiest example of a quantum probability space: Let
A = M2 (C) be the W*algcbra of all complex 2 x 2 matrices a = (a;k ) and choose A E] 1/2, 1] . Then T.A (a) = >au + ( 1  >.)a22 defines a normal state on A. As in the case of classical probability we will often write IE ,\ (a) in place of a.
T.A (a) , and we call it the expectation of Now consider the following model of tossing a fair coin: choose Ao as the commutative W*algebra C { O . l } and let p,o : Ao + C be given by p,o(f) = � (! ( 0) + f ( 1)). The underlying classical probability space is to be rediscovered in Ao as the set of two idempotents 1 { o } and 1 { 1 } where 1 A denotes the indicator function of a subset A of a given set. Ao is spanned also by the elements 1r
We now construct all injective *algebra homomorphisms : Ao + A satisfy ing T,\ o = fLO · In the sense of section 13.2 this means that we will find all abstract probability spaces (B, T,\ 1 13 ) which arc equivalent to (Ao, p,o). An easy calculation shows that they are given by := with 1r
1r
1rz

z
1
) = : Qz
13.3.
Quantum Bernoulli experiments
and linear extension, where Pz  Q z =
( � 0 ) = : Xz .
z E lC
193
is a parameter with l z l
= 1. n2 (X) =
In summary we have seen that the simplest quantum probability space (A. T.\) contains a oneparameter family (Az , T.\ IAJ of models of classical coin tossing. For w = eis, z = e it ( n < s, t :.  1) sin(s  t). The absolute value of it attains its maximum at  t = ±n /2. The pair (Xw , X2) with z =  iw is called a quantum coin tossing. It is unique up to automorphisms. More precisely this means the following: choose w = e it (t E]  n, n] ) , and consider the automorphism 1.{) on A given by I.{)( a) = u*au where 0 e it/2 u. 0 e it/2 Then i.p(Xl ) = Xw ' i.p(X i ) = x iw and T.\ 1.{) = T.\ . So in the following we need only to consider the quantum coin tossing (X1 , X_ i ) . Notice that X1 = ax , X_i = ay and [Xw , X_,w]x = 2ia2 , where ax , a y, az denote the Pauli matrices. The following important matrices can be constructed by (ax , ay) : 1 0 1 . (13.3. 1 ) (ax + wy ) , b2 0 0 1 0 0 . (13.3. 2) b+ 2 (ax  wy ) , 1 0 1 0 0 ( 13.3.3) 1  [ax , ay] · bo 0 1 2i Obviously the following formulae hold: s
_
(
( ( (
0
)
) ) )
ax = b+ + b  , ay = i · (b+  b  ) , [b  , b+] = a2 = 1  2b0 , Xw = wb+ + wb  .
These formulae show that the random variables b+ and b are in a certain sense more basic than Xw though they are not selfadjoint. As in the classical case we describe the experiment of n ( E N) independent coin tossings by the nfold tensor product (A®n , T�n) . The algebra correspond ing to the kth trial is described by the elements 1 129 · · • 129 1 ®x 129 1 129 · · · 129 1 = : X k 'v" (k 1 ) times ( n k) times
'v"
194
13.
Quantum Bernoulli Experiments
where x E A is arbitrary. Describing k times repetition of our quantum coin tossing we obtain Sw. k := L;= l Xw1 . We have k [Sw ,k, s iw, k] = 2i k·
La
J =l
z,
The pair (Sw, k l S iw, k ) is called a quantum Bernoulli experiment. Setting Ak := A0 k ® 1 ® · · · ® 1 = {a ® 1 ® · · · ® 1 : a E A0 k } 'v" 'v" n  k times n  k times we obtain a natural filtering lC · 1 C A1 C · · · C An = A0n . Moreover there also exists a conditional expectation !Ek from An onto Ak which is given by IEk (a ® b) = T; ( n k ) (b) · a , where b E A0 ( n k ) , and linear extension.
13.4
The internal quantum processes
Now we consider a polysaturated model *V(JR:.) of the full structure V(JR:.) over JR:.. As usual we replace the standard integer above by an infinitely large integer N. Then all considerations in the previous section remain true for internal objects. We use the discrete time interval T = { � : 0 ,N (1/N) =
1/N.
The socalled internal Itotable for the increments is easily computed:
a• a+o a§. a;,
pa2:a: I a0; I a0� I p2aa:;; p2 a; 0 0 p2 a� 0 a+§. a§.o 0 0 a• a;, 0
ptv =
A� + lvi B",t. + l v l 2 A; lvl
s
s
§.
Let 0 # v E *IC be arbitrary. Then the process (P.�_) given by is called the Poisson Process with parameter v . Now we compute the characteristic operator functions of these pro cesses. Whenever V = (ifi)t_ is a stochastic process its characteristic oper ator function is defined by fv (u, t) = exp( i uvt), where u E *IF!:.. Obviously A ( t ) l u= O = iifi and fv ( u, t) is unitary iff Vi is selfadjoint . The processes V = (Vi) we considered so far are all of the form u,
Vi =
I: d§.,
O:S;§.
(1 3 . 5 . 6 )
'"
(13.5. 7)
13.5. From the internal to the standard world
199
Consider now the projection IE.t. from 7{ onto 1l.t. which is the Hilbert space coming from A..t. (see section 13.3 ) . IE..t. is given by lEt ( ew ) Then we obtain IE_.t.A�
=
{ ew 0
w (�) = 0 for all � 2> t
else
A�IE..t. for each rt E family (A�h< 1 is ada�ted t� (H..t.h < 1 ·
{ • , o, + ,  }
=
or in other words the
We prove now some continuity properties. To this end we introduce 7{ ( n)
:=
{ L Yw ew : Yw E *lC} l w l= n
and we set 7{ : = 7{TN for short. Then 7{ = 1_;{= 0 7{ (n) where _i denotes the orthogonal direct sum. More over A; (H( n) ) C 7{ ( n) Ai (H( n) ) C 7{ ( n) Ai (H( n ) ) C 7{n + 1 , and finally Ai (H( ;;:) ) C 7{ ( n1 ) where 7f ( 1 ) := {0 } = : 7{ ( N+ 1 ) . For y = L l w l =n Yw ew and 0 :S §. < t we obtain ,
,
L I� L
II A� y  A� y ll 2
lw l =n
§."5cy.) (>.: Lebesgue measure) the tensor n
:F
=
product H® is nothing else than L2 ( [0, 1] 71 , .\71 ) where A 71 denotes the dimensional Lebesgue measure. and the projection Pn is given by
n
Now we give the more or less classical definition of the basis quantum processes. To this end we set
h o · · · fn : = Pn ( h Q9 · · · � fn ) · 0
For
0 :S t
h : w(�) = 1). Then I(w) := (h (w) , . . . , �n (w)) E *Xn n Tn . Moreover if f := (kl /N. . . . , kn/N) E *Xn n Tn then f = f( w) for w = 1 M where M = {h (w) , . . . , �n (w) }. We consider the dense subspace Cn of L2 (Xn , An) consisting of the re strictions to Xn of continuous functions on the closure Xn . For E Cn we set S(f) (w) = * (f(w)) . This is an element of L2 (D n , mn ) satisfying l l f l l 2 :::::J II S(f) ll 2 · It follows that Cn is linearly and isometrically embedded into L2(D n , LmJ , and this embedding can obviously be extended to the whole of L2 (Xn , An) . Since L2 (D n , Lmn ) is linearly and isometrically embedded in Ln , we obtain that the symmetric tensor product L2 ( [0, l] , A) O n can be em bedded in Ln . An obvious extension then yields an embedding denoted by U of the symmetric Fock space F+ ( L2 ( [0 1 ] , A) into £. Call y the image of this embedding and set Yo = G n K0 .
fd
f
f
f n!
g n! >
t1
f
f
,
2 Yo is dense in y. Moreover the restrictions of A� to Yo yield closed densely defined operators which are selfadjoint for � E { •} and adjoint to each other in the other cases and which moreover satisfy
Theorem
on u 1 (Qo) where to (13. 6.!J
o,
A�
aTe the operators given by equations
(13. 6. 1)
'UP
The proof is obvious.
References [ 1 ] R. M . ANDERSON, "A nonstandard representation for Brownian motion and Ito integral", Israel .J. Mathern. , 25 1546. [2] S . ATTAL, Approximating the Fock space with the toy Fockspace, in .J. Azema, I\1 . Emery, M. Ledoux, I\L Yor (eds.), XXXVI, Lecture Notes in Mathematics 180L SpringerVerlag, Berlin New York, 2003.
Seminaire de Probabilites
References
[3] A.
GUICHARDET,
205
Symmetric Hilbert spaces and related topics, Lecture
Notes in Mathematics 26L SpringerVerlag, Berlin, 1972.
[4] R.
L . HUDSON and J . M. LINDSAY, "A noncommutative martingale rep resentation theorem for nonFock Brownian motion", .T . Funct. Anal., 6 1
(1985) 202221 .
[5] R.
R . PARTHASARATHY , "Quantum Ito ' s formula and stochastic evolutions", Comm. Math. Physics, 93 (1 984) 311323. L. HUDSON and K .
[6] R.
L. HUDSON and R. F. STREATER, "Ito ' s formula is the chain rule with Wick ordering", Phys. Letters, 86 ( 1982) 277279.
[7]
"Structure des cocycles markoviens sur respace de Fock", Prob. Theory Re. Fields, 75 (1987) 291 316.
.T . L . .T OURNE.
[8] B .
KDMMERER and H. MAASSEN , "Elements of quantum probability", Quantum Probab. Communications, QPPQ, 10 (1998) 73100.
[9]
H . MAASSEN, Quantum Markov processes on Fock space described by integral kernels, in L. Accardi, W. von Waldenfels, Lecture Notes in Math ematics 1136, SpringerVerlag, Berlin. New York, 1985.
Quantum probability and applications II, Proceedings Heidelberg 1984, [10] M. LEITZMARTINI. Quantum stochastic calculus using infinitesimals, Ph.D. Thesis, University of Tubingen, 2001 . [11] P . A . MEYER. Quantum probability for probabilists, Lecture Notes Mathematics 1 538, SpringerVerlag, Berlin, New York, 1993.
[12]
m
K. R. PARTHASARATHY, The passage from random walk to diffusion in quantum probability, (1988) , Ap plied Probability Trust, Sheffield, 231  245.
Celebration Volume in Appl. Probability [13] K. R. PARTHASARATHY, An introduction to quantum stochastic calculus, Birkhauser. Basel, 1992.
Applications of rich measure spaces formed from nonstandard models Peter Loeb*
Abstract \i'Ve review some recent work by Yeneng Sun and the author. Sun'i:l work shows that there
are
results, some used for decades without a rigourous
that arc only true for spaces with tho rich structmo of Loeb measure spacel:l. His j oint work with the author uses that structure to extend an important result on the purification of measure valued maps. foundat.ion,
14. 1 In
Introduction 1975 [ 1 1] ,
the author constructed a class of standard measure spaces
fanned on nonstandard models.
These spaces, nmv called "Loeb spaces" in
the literature, arc very close to the underlying internal spaces, and ru·c rich in
s t ruc t ure.
(Sec
1 121 by Yeneng Sun
the author's throe clmptcrs and Osswald's two chapters in
for background. ) 'vVe will briefly review here some recent work
that uses these measure spaces. Sun has shown that there are results, some
used in applications for decades without a rigorous foundation, that are only true for spaces with the rich structure of Loeb measure spaces. We will fol l ow the description of Sun'::; work with a detailed proof of a special case of t.hc recent
result in
[H] by Yeneng Sun and the author on A counterexample shows that the
valued maps.
the purification of measure
result we give is false if the
Loeb space we use is replaced by the unit interval with Lebesgue measure. We
note in passing hero that the result in
[13]
by Osswald, Sun, Zhang and the
author presents yet another example needing rich measure ::;paces.
*
Department
of
Mathematics, 'University
loeb@math. uiuc . edu
of
Illinois, Urbana, IL 61801.
14.2. Recent work of Yeneng Sun
14.2
207
Recent work of Yeneng Sun
We begin with some work in the late 1990's of Yeneng Sun ( [16] , [17] , [18] , and Chapter 7 of [12] ) . That work, Sun's alone, is published elsewhere, so what is said here is an invitation to read, and not a substitute for, the original articles. To set the background we consider the following question: Does it make sense to speak of an infinite number of independent individuals or random variables indexed by points in a uniform probability space? Can one, for example, reasonably consider an infinite number of independent tosses of a fair coin with the tosses indexed by a uniform probability space, and if so, does it make sense to say that half the tosses should be heads? Of course, there is no problem in speaking of independent random variables indexed by the first integers with each integer having probability 1/ An infinite index set with a uniform probability measure, however, must be uncountable; there is no uniform probability measure on the full set of natural numbers. The problem thus evolves to finding the possible meaning of an uncountable family of independent random variables. Whatever way one approaches this problem, the usual measuretheoretic tools fail. A natural attempt to generalize coin tossing replaces the natural numbers with points in the interval [0, 1], and thus replaces sequences of 1 ' s and  1 ' s with functions from [0, 1] to the twopoint set { 1, 1 } . This is a reasonable model for an uncountable family of independent coin tosses. In 1937, Doob [2] exhibited a problem with this and similar spaces of functions when standard techniques arc applied. To sec the problem, let denote the set of {  1, 1 } valued functions on [0, 1] , and let be the product measure on constructed from the measure taking the values 1 / 2 at 1 and 1/2 at  1 . In the usual construction of an appropriate aalgebra on measurable sets are formed from the algebra of cylinder sets, with functions in each cylinder set restricted at only a finite number of elements of [0, 1] to take either the value 1 or  1 . It follows that each measurable set in is determined in a way described below by a countable subset of [0, 1 ] , and so the following result holds.
n
n.
P
D
D
D,
D Proposition 14.2.1 Fix any h E D. Set Mh := {w E D : w(t) = h(t) except for countably many t E [0, 1 ] } . Then Mh has Pouter measure 1 . Proof. For each measurable B � D, there is a countable set C [0, 1] such that for all o:, (3 in D, if o:(t) = (3 (t) for all t E C, then E B if and only if (3 E B. Suppose Mh � B . Given w E D, Let w' agree with w on C and agree with h on [0, 1] \ C. Then w' E Mh, so w' E B. It then follows that w E B . Thus B = D, so the outer measure P*( Mh ) = 1 . a
C
D
14. Applications of rich measure spaces
208
Remark 14.2.2 Note that if w E lvh, then since Lebesgue measure A of a countable set is 0, w = Aa.e. on [0, 1 ] . This is true if is nonmeasurable, or if 1, or if  1 . Now outer measure when applied to the intersection of measurable sets with Mh is finitely additive, hence countably additive. Since the outer measure of Mh is 1 , one can trivially extend P to a measure P with P(Mh) = 1 . Thus, no matter what might be, one can claim that P almost every function is equal to at Aalmost very point of [0, 1]. This, and other highly questionable arguments have been used for decades to work around the measure theoretic problem indicated by Doob's example.
h
=
h
=
h
h
h
h
Another approach to representing a continuum of independent random vari ables is to consider a function w , called a process, where is an index from an uncountable probability space called the parameter space (the probability measure need not be uniform) and w is taken from a second probability space called the sample space. The question then is whether it makes sense to work with the usual product of these two probability spaces. Sun has shown in Proposition 7.33 of [12] that no matter what kind of measure spaces, even Loeb measure spaces, one might take as the parameter space and sample space of a process, independence and joint measurability with respect to the classical measuretheoretic product, i.e. , formed using measurable rectangles as in [15] , are never compatible with each other except for a trivial case. Here is the exact statement of that proposition.
f(i, )
i
Let (I, I, ) and (X, X, v) be any two probability spaces. Form the classical, complete product probability space (I x X, I 129 X, 129 v ) . Let f be a function from I x X to a separable metric space. If f is jointly measurable on the product probability space, and for 11almost all (i 1 , i 2 ) E !xi, h and fi2 are independent (call this almost sure pairwise in dependence), then, for ttalmost all i E I, f( i. ) is a constant f1m ction on X. Proposition 14.2.3 (Sun)
f1
f1
f1 129
·
Sun notes that Proposition 14.2.3 is still valid when f1 has an atom A . The almost sure pairwise independence condition implies the essential constancy of the random variables fi for almost all i E A. In the articles cited at the beginning of this section, Sun has shown that a construction overcoming these measuretheoretic problems is obtained by forming the internal product of internal factors, and then taking not just the Loeb space of each factor, but also the Loeb space of the internal product. This yields a rich extension of the usual product aalgebra formed from the Loeb factors, one that still has the Fubini Property equating the integral over the product space to the iterated integrals over the factor spaces forming that product. Here in more detail is that construction.
14.2. Recent work of Yeneng Sun
209
Sun's starts with internal spaces ( T, T, A) and ( D, A, P) . The space T may be a hyperfinite set with A given by uniform weights. He then forms the Loeb spaces (T, L>. (T) ) ) and (D, L!L (A) , P ) . He lets A Q9 P denote the internal product measure, while T Q9 A denotes the internal product aalgebra, and L>. (T) Q9 Lp (A) denotes the classical product aalgebra formed from L>. (T) and Lp (A) as in [15] . On the other hand, forming the Loeb space from the internal product T Q9 A produces a larger aalgebra L>.0P (T Q9 A) on T n . ( Sec the following propositions. ) Here arc some properties of what we shall call the big product space (T x D, L>.0p (T Q9 A) , �) . X
The big product space depends only on the Loeb factor spaces (T, L>. (T) , >:) and (D, Lp (A) , P) . Proposition 14.2.5 (Anderson [1] ) If E E L>. (T) Q9 Lp (A) , then E E L>.0p (T � A) and � (E) = ); ® P(E) . Proposition 14.2.6 (Hoover (first example) , Sun [ 1 7] ) The inclusion of L>. (T) Q9 Lp (A) 'in L>.0P (T Q9 A) is strict if and only if both ); and P have nonatomic parts. Proposition 14.2. 7 (Keisler [6] ) A Fubini Theorem holds for the big prod uct space, Proposition 14.2.4 (KeislerSun [7] )
In his articles, cited above, Sun shows that to work with independence in a continuum setting, as has been attempted in informal mathematical applica tions without rigor, and for decades, one needs a rich product aalgebra such as the aalgebra of a big product space. Here is that general result as stated in Proposition 7.4.1 of [12] and proved in Theorem 6.2 of [17] .
Let X be a complete, separable and metrizable topological space. Let M (X) be the space of Borel probability measures on X, wher·e M(X) is endowed with the topology of weak convergence of measures, Let be any Borel probability measure on the space M(X) . If both ); and P are atomless, then there is a process f from (T n , L>.0P (T Q9A), GP) to X such that the random variables ft = f ( t, ) are almost surely pairwise independent (i,e,, for �almost all (t1 , t2 ) E T x T, fh and ft2 are independent ), and the probability measure on M(X) induced by the function Pft 1 from T to M (X) is the given measure Remark 14.2.9 Here, for any given t E T, P ft 1 is the probability measure on X induced by the random variable ft D + X. It is the measurable mapping from T to M (X) taking the value P ft 1 at each t E T that induces Proposition 14.2.8 (Sun) fL
·
X
fL .
:
the measure fL on M (X).
14. Applications of rich measure spaces
210 Remark 14.2.10
(
A consequence of the proposition is that when the space
T. L>.. (T), :\) is a hyperfinite Loeb counting probability space and 'ji is a non
atomic Loeb probability measure, it makes sense to use the big product space as the underlying product space for an infinite number of equally weighted, independent random variables or agents. As part of his work with large product spaces, Sun has extended the law of large numbers. The usual strong law states that if random variables Xi , E N, are independent with the same distribution and finite mean m, then � L:i= l Xi tends almost surely to the constant random variable m. That is, for almost all samples the value of the sequence at tends to the constant m.
i
w, w Theorem 14.2. 1 1 (Sun) Let f be a realvalued integrable process on the big product space. If the random variables ft := f(t, ·) are almost surely pair wise independent, then for almost all samples w E D, the mean of the sample function fw := f(·,w) on the parameter space T is the mean off viewed as a random variable on the big product space. There is no requirement of identi cal distributions. Another facet of this same work deals with independence. It is well known that for a finite collection of random variables, pairwise independence is strictly weaker than mutual independence. Sun has shown that for processes on big product spaces, pairwise independent and mutual independent coalesce. and they coalesce with other notions of independence that are distinct for a finite number of random variables. This implies asymptotic results for finite families of random variables where the families are ordered by containment and have increasing cardinality.
14.3
Purification of measurevalued maps
We now turn to a special case of the joint work of Yeneng Sun with the author in [14] . That special case generalizes the following celebrated theorem of Dvoretzky, Wald and Wolfowitz (see [3] , [4] , [5] ) .
Let A be a finite set and M(A) the space of probability measures on A. Let (T, T) be a measurable space, and ILk ! k = 1 , · · · , finite, atomless signed measures on (T, T) . Given f : T + M (A) so that for each a E A, f( ·) ( {a}) is Tmeasurable, there is aTmeasurable map g : T A so that for each a E A, and each k l f(t)({a})dfLk (t) = fLk ({t E T : g(t) = a}). Theorem 14.3.1
m,
+
: A so that for each l :::.; lm and each j :::.; N, 9m ( t) = a[ on T/'m . For each continuous, realvalued 7/J on A , for each m _2: 1 and each k E K,
lm N
= L L ( 7/J (a [ )rj (A[ ) ) f1 k (Sj ) l=l J = l
lm N
( )
= L L 7/J (a[ ) !1 k T/'m l=l j = l
approximates as m > oo the integral
£ (l >/� (a)f(t) (da) ) Mk (dt)
�
·
·
t, £ (1r t, � 1, (1r
)
>,b (a) j(t) (da) �k (dt)
)
>,b(a) o, (da) Mk (dt ;
215
References so
1/J (a[ ) !Lk (Tzj,m ) . L (1 1/J (a)j(t)(da) ) fLk (dt) It follows from the lemma that there is a Tmeasurable mapping g from T to lm N =LL l=l j = l
A such that for each k E K and for each continuous realvalued function e on A from a countable dense set of such functions, and therefore for each bounded Borel measurable function e on A,
L B(g(t))tLk (dt) L l B(a)j(t)(da) fLk (dt). =
D
References [1] R. M . ANDERSON, "A nonstandard representation of Brownian motion and Ito integration", Israel J . Math . , 25 (1976) 1546. [2] .T . L . D ooB, " Stochastic processes depending on a continuous parameter", Trans. Amer. Math. Soc., 42 (1937) 107 140. [3] A. DVORETSKY, A. WALD and .J . WoLFOWITZ, "Elimination of random ization in certain problems of statistics and of the theory of games", Proc. Nat . Acad. Sci. USA, 36 (1950) 256260.
[4] A. DVORETSKY, A. WALD and .J . WoLFOWITZ, "Relations among certain ranges of vector measures", Pac . .J. Math. , 1 (1951) 59 74. [5] A . DVORETSKY, A . WALD and .J . WoLFOWITZ, ""Elimination of ran domization in certain statistical decision problems in certain statistical decision procedures and zerosum twoperson games", Ann. Math. Stat . , 22 (1951) 1 21.
[6] H . .J . KEISLER,
An infinitesimal approach to stochastic analysis, Memoirs
Amer. Math. Soc.
48, 1984.
[7] H .. T . KEISLER and Y . N . SUN, "A metric on probabilities, and products of Loeb spaces", .Jour. London Math. Soc . , 69 (2004) 258272.
14. Applications of rich measure spaces
216
[8] 1\ I . A . K HAN , K . P . RATH and Y. N. SuN. "The DvoretzkyWald Wolfowitz theorem and purification in atomless finiteaction games", In ternational Journal of Game Theory, 34 (2006) 91  104.
[9] M . A . KHAN and Y. N . Su N , "Noncooperative games on hyperfinite Loeb spaces", J. Math. Econ. , 31 (1999) 455492. [10] P . A . LOEB, "A combinatorial analog of Lyapunov's Theorem for infinites imally generated atomic vector measures". Proc. Amer. Math. Soc . , 39
(1973) 585586. [ 1 1] P . A . L OEB, " Conversion from nonstandard to standard measure spaces and applications in probability theory", Trans. Amcr. Math. Soc., 2 1 1
(1975) 1 1 3 122.
Nonstandard Analysis for the Working Mathematician. Kluwer Academic Publishers, Amsterdam, 2000.
[12] P . A . LOEB and M. WoLFF, cds.,
[13] P . A . L OEB, H . OsswALD, Y. SuN and Z . ZHANG, "Uncorrclatcdncss and orthogonality for vectorvalued processes", Tran. Amer. Math. Soc . , 356
(2004) 3209 3225. [14] P . A . LOEB and Y. Su N , "Purification of measurevalued maps", Doob Memorial Volume of the Illinois Journal of Mathematics, 50 (2006) 747 762. [15] H . L . RoYDEN,
Real Analyszs, third edition. Macmillan, New York, 1988.
[16] Y. SuN. "Hyperfinite law of large numbers", Bull. Symbolic Logic, 2 (1996) 189198. [17] Y. Su N , "A theory of hyperfinite processes: the complete removal of in dividual uncertainty via exact LLN". J. Math. Econ. , 29 (1998) 419503. [18] Y. SU N , "The almost equivalence of pairwise and mutual independence and the duality with exchangeability", Probab. Theory and Relat. Fields, 1 1 2 (1998) 425456.
More on Smeasures David A. Ross
15.1
*
Introduction
In their important. (but. often overlooked ) paper [1] . C. Ward Henson and Frank \iVattenberg introduced the notion of Smeasura.biz.ity, and showed that Smeasurable functions are "approximately standarcP' ( in a sense made precise in the next section) . In a recent paper ([3]), the author used this machinery to transform a well known nonstandard plausibility argument for the RadonNikodym theorem into a correct and complete nonstandard proof of the theorem. The problem of finding an "essentially nonstandard" proof for RadonNikodym had been one of long standing. A ltho ugh Luxemburg gave a nonstandard proof as long ago as 1972 ([2] ) , he obtained the result as a consequence of another equally deep theorem in analysis clue to Riesz. ( Beate Zimmer [6] has recently proved vectorvalued extensions of RadonNikoclym starting from the same plausibility argument, though using different standardizing machinery than that in [3] or the present paper. ) In this paper I use an Smeasure argument very like the one in [3J to give an intuitive nonstandard proof of the Riesz result used by Luxemburg. Along the way [ gi ve a new proof for the main technical result from [1] on Smeasures, a uew noustaudard proof for Egoroff's Theorem, and a nonst. 0 3A E A fL(A) > fL( X )  c & fn + f uniformly on A.
Theorem 3
222
15. More on Smeasures
Proof. For n E N put 9n = inf m> n fm and h n = supm > n fm · Note that for almost all x, hn (x)  gn (x) is nonnegative and nondecreasing in n with limn_, = hn(x)  9n(x) = 0. Put ?Jn = 0 *gn , h n = 0 *h n . Let E = {x E *X : limn _,= hn(x)  ?Jn (x) c/c 0}. Note E E As, so fLL(E) = p,(S(E)) = p,(0) = 0. It follows that for some A E A, *A c;;;; E C and p,(A) > p,(X)  c . It remains to show that fn converges to f uniformly on A; equivalently, that hn  9n converges to 0 uniformly on A. Fix 6 > 0. For x E *A there is a k E N such that hk(x)  gk(x) < 6. Note *h k( x )  *g k (x) < 6 as well. Let ¢(x) be the least k such that *h k (x)  *g k (x) < 6. The function ¢ : *A + N is internal and finitevalued, so has a standard upper bound N E N. Then *h k  *g k < 6 on *A for every k 2 N, so h k  9k < 6 on A; this completes the proof. D
15.4
A Theorem of Riesz
Let T be a continuous linear real functional on .C 2 (X, A, fL ); then there is a g such that for every f E .C 2 (X, A, p,), T(f) = J fg dp,.
Theorem 4
g is actually in .C 2 (X, A, p,) . By saturation let A be a *finite algebra with A0 c;;;; A c;;;; *A . There
After the proof we show that the function
Proof. is an internal *partition II of *X which corresponds to A in the sense that the latter is the internal closure of the former under hyperfinite unions. Now, define
i (x)
:=
{
*T( xv ) *p, (p )
0,
'
x E p E II and *p, (p )
>
otherwise.
Let a : II + *X be an internal choice function, that is, For any bounded, measurable f : X + rn:.,
T (f) = L * T (*f xv ) p p p
0;
ap E p for p E II. ( 1 5.4. 1 ) ( 1 5.4.2) ( 1 5.4.3) (1 5.4.4)
223
1 5 .4. A Theorem of Riesz
1p � L 1 *j(x yY (x) *df1 p p = J *f :Y *d/1 =
L *f(avYf (x) *d/1
(1 5.4. 5)
p
( 1 5.4. 6) ( 1 5.4. 7)
The steps from ( 1 5.4. 1 ) to ( 15.4.2) and (15.4.5) to ( 1 5.4.6) follow from boundedness of f and the definition of A. For (15.4.2) to ( 1 5 . 4.3) note that by continuity of T, if p E II and *!1(P ) = 0 then *T( xv ) = 0. Moreover, if *f is replaced in the above by any internal function h : *X + *JR:. with the property that h is constant on each p E II, then all but the first equality in the above still holds, and we obtain *T(h) = J hi d*f1 . In particular, if h = XA for some A E A then *T( x A ) = fA :Y d*f1. The proof now proceeds as follows : (i) Show :Y is finite almost every where (and therefore 1 =0 :Y exists almost everywhere) . (ii) Show that :Y is Sintegrable (and therefore 1 =0 :Y is integrable) . (iii) Put G = IE[riAs] (the conditional expectation of 1 ) . (iv) Let g be the restriction of G to X; by Theorem 2 g is Ameasurable. (v) Show that g works. For (i) , write [:Y < n] = {x E *X : :Y(x) < n } ( n E *N) , and [i < oo] = U n EN [:Y < n] . Suppose (for a contradiction) that f1L ( [:Y < oo] ) < 1  r for some standard r > 0. Then !1 ( [:Y < n ] ) < 1  r for each standard n E N, so !1 ( [:Y < H] ) < 1  r for some infinite H. But then oo > T(1) � *J1:Yd*f1 (by the note above) 2 *J [:Y ? H] :Yd*/1 2 H*!1 ( [:Y 2 H] ) > rH, which is infinite, a contradiction. For (ii) , the reader is referred to [4] for a discussion of Sintegrability. In particular, to show that :Y is Sintegrable, it suffices to show that VH in finite, *J :Y > H] :Yd*/1 � 0. (Indeed, this can be adopted as the definition of I Sintegrability. ) So, fix such an H. Note [:Y > H] E A; it follows that * J :Y > H] :Yd*/1 = I *T(X[:Y > HJ ) · Suppose (for a contradiction) that *T(X[:Y > HJ ) > r > 0, r stan dard. By (i) , *!1 ( [:Y > H] ) � 0. It follows from transfer that for every stan dard n E N there is a Bn E A with T(XBn ) > r and f1(Bn) < 1 / 2 n . Put B n := Um > n Bm . Then 11(B n ) < L m > n 2  m = 2  n + l , but T(XBn ) > r. As n + oo, X;;n + 0 in £ 2 but T(XBn ) > :;., contradicting continuity of T. It follows from Sintegrability of i that 1 =0 :Y is integrable, so its con ditional expectation with respect to As, IE[riAs] , exists; put G = IE[riAs] . (Conditional expectation is defined in the next section, where a simple non standard proof for its existence is given. The reader is referred to chapter 9 of [5] for properties of the conditional expectation.)
224
1 5 . More on Smeasures
Finally, let g G l x , which ( as noted above ) is Ameasurable by Theorem 2. It remains to show that g satisfies the conclusion of Theorem 4. Let f E J:} (X, A, p,) , and suppose first that f is bounded. This ensures that (*!)1 is Sintegrable, and (0*j)r E J:} (*X, AL, p,L) . Then =
J *FI d *p, � J (0*!) { dp,L = j IE W *f)r i A s ] dp,L j C *f) IE [riAs] dp,L = J (0*j)G dp,L = J fg dp,
T(j) �
=
( 1 5.4.8) ( 1 5.4.9) ( 15.4. 1 0) ( 15.4. 1 1 ) (15.4. 12) (15.4. 1 3)
The step from ( 15.4.8 ) to ( 15.4.9 ) is by Sintegrability, ( 1 5.4. 10 ) to ( 15.4. 1 1 ) is a standard property of conditional expectation ( using the fact that 0*j is Asmeasurable ) , and (15.4.12 ) to ( 1 5.4. 13 ) is Corollary 1 with p = 1 . For unbounded f E J:} (X, A, p, ) and n E N, let fn = max { n, min { !, n } } . By the result above , T(jn ) = J fn g dp,. By continuity of T and Lebesgue's Dominated Convergence Theorem ([ 5 ] , Theorem 5.9 ) , T(j) = J fg dp,. D The Riesz Theorem is usually stated in the following nominally stronger form.
Let T be a continuous linear real functional on .C 2 (X, A, p,) ; then there is a g E .C 2 (X, A, p,) such that for every f E .C 2 (X, A, p,), T(j) = J fg dp,.
Corollary 2
Proof. It suffices to show that any g satisfying the conclusion of Theorem 4 is already in .C 2 (X, A, p,). Since T is continuous, it is bounded, so there is a constant C such that for any f E .C 2 ( X, A, p,) , T(j) � CIIJII 2 · Put 9n = min { g, n } . Then ll9n ll� = Jg� dp, S fgn g dp, = T(gn ) S Cll9n l l 2 · Divide both sides of this inequality by ll9n ll 2 and square, obtain J g� dp, ll9nll � S C 2 . The result now follows by Fatou's Lemma ([5] , Lemma 5.4) . D
15.4. A Theorem of Riesz 1 5 .4 . 1
225
Conditional expectation
Theorem 5 Suppose (X, A, fL) zs a probabdzty measure, that 'B C:: A is another aalgebra, and that j E ,.C} (X, A, tt) . There is a 'B measurable function g : X ___, ffi. such that for any B E 'B, f dfL 9 dfL.
JB
=
JB
The function 9 is called the conditwnal expectation of f on 'B, and denoted by IE [f i'B] . To avoid circularity, it is necessary to show that IE [f i 'B] exists without use of the Riesz Theorem (or related results, such as the RadonNikodym Theorem) . Such a proof appears in [3] . This section presents a modification of that proof which is even more elementary, in that it does not require the Hahn decomposition. Without loss of generality f 2 0. Put 9 {9 E ..CJ (X, 'B, fL) : VB E 'B , 9 dfL r  l/2 n ; we may assume that 0 m Bn , and put (j  g)dfL s
CX)
Jc\ Bx u
r u 9') dfL = r
Jc
JB XB
_
9) dfL +
.fCnB=u  g) dfL .
15. More on Smeasures
226
The first term in this sum is nonnegative since g E 9 . The second is nonnegative since otherwise g')dfL 2 0, so g' E fj )dfL > s. It follows that 9 . Since s > 0, !L(BcxJ) > 0, so g' dfL g dfL + 6fL(B00) > r. a contradiction.
JBx\C ( j 
J
=
J
Jc ( J 
References [1] C . WARD HENSON and FRANK WATTENBERG , "Egoroff ' s theorem and the distribution of standard points in a nonstandard model", Proc. Amer. Math. Soc., 81 ( 1981) 455461 . [2] W . A . J . LUXEMBURG, O n some concurrent binary relations occurring in analysis, in Contributions to nonstandard analysis (Sympos., Oberwol fach, 1970), Studies in Logic and Found. Math. , Vol. 69, NorthHolland, Amsterdam. 1972. [3] DAVID A. Ross, Nonstandard measure constructions  solutions and problems, in Nonstandard methods and applications in mathematics, Lec ture Notes in Logic, 25, A.K. Peters, 2006. [4] DAVID A . Ross. Loeb measure and probability, in Nonstandard analysis (Edmburgh, 1996), NATO Adv. Sci. Inst. Ser. C. Math. Phys. Sci., vol. 493, Kluwer Acad. Publ., Dordrecht, 1997. [5] DAVID WILLIAMS, Probability with martingales, Cambridge Mathematical Textbooks, Cambridge University Press, Cambridge, 1991. [6] B EATE ZIMMER. "A unifying RadonNikodym theorem through nonstan dard hulls", Illinois .Journal of Mathematics, 49 (2005) 873883.
A RadonNikodym theorem for a vectorvalued reference measure G. Beate Zimmer*
The conclusion of be represented
a
Abstract
RadonNikodym theorem iH that
as �m
that for all measurable sets Lebesgue)
a
measure f1. can
integral with respect to a reference measure
A, p(A)
such
JA .f1,. (x) d).. with a (Bochner or
=
integrable derivative or density .fw The mcasme ).. is usually a
countably additive <Jfinit.e measure on the given measure space and the
mcasmc JL is absolutely continuous with respect to ).. . Different theorems
have different range spaces for p., which could be the real numbers, or Banach s paces with or without the RadouNikodym property.
Iu this
paper we ge1ieralize to derivatives of vector valued measures with re spect a vectorvalued reference measure. theorem for vector
measures
Vvc present a R.adonNikodym
of bounded variation that arc absolutely
continuous with respect to another vector measme of bounded variation. W'hile it is easy in settings sud1 on the interval
[0, 1]
as
Jl <
fiu ( *E) by
t.p1, (:1:) 2:!1 :���:\ lA, (.1; ) , a vector valued reference measure does uot. =
allow this approach.
as
the quotient of
two vectors
in different B anach
spaces is undefined. Furthermore, generalizing to a vector valued control mcmmre uecessita(;es the usc of a generalization of the Bartle
bilinem· vector integral.
16 . 1
integral , a
Introduction and notation
For noustandard notions and notatious not defined here we refer for exam
ple to the book by Albeverio, Fenstad, 1IoeghKrohn and Lindstrom
t.he survey on nonstandard hulls by Henson and 1\tloore *Department of Tvfathematics and Statistics, Texas Corpus
Christi, TX 781112.
[1]
and
[6] or the introduction
A&M Universit.y 
Corpus Christi,
16. A RadonNikodym theorem
228
to nonstandard analysis in [7] . Plenty of information about vector measures can be found in [5] . Throughout, D is a set and I: is a aalgebra of subsets of D . E, F and G denote Banach spaces. fL : I: + E and v : I: + F are countably additive vector measures of bounded variation. The variation of v is defined as I vI (A) = sup L 1rEII llv(A") II where A E I: and II is a finite partition of A into sets in I: . B y Proposition I . l . 9 i n [5] the variation lvl of a countably additive measure v is also a countably additive measure on I:. We think of the nonstandard model in terms of superstructures V(X) and V(*X) connected by the monomorphism * : V(X) + V(*X) and call an ele ment b E V(*X) internal, if it is an element of a standard entity, i.e. if there is an a E V (X) with b E *a. We assume that the nonstandard model is at least Nsaturated, where N is an uncountable cardinal number such that the cardinality of the aalgebra I: of I vimeasurable subsets of D is less than N. Let (*D, Llvi (*I: ) , lvl ) denote the Loeb space constructed from the nonstan dard extension of the measure space (D, I:, I v i ). This measure space is obtained by extending the measure 0*1vl from *I: to the aalgebra generated by *I:. _I he completion of this aalgebra is denoted by Llvi (*I: ) . The Loeb measure l v l is a standard countably additive measure. The functions we work with take their values in a Banach space E or its nonstandard hull E. The nonstandard hull of a Banach space E is defined as E = fin(* E)/ the quotient of the clements of bounded norm by clements of infinitesimal norm. The nonstandard hull is a standard Banach space and contains E as a subspace. We denote the quotient map from fin(*E) onto E by or 'if£ . Whenever we encounter products of Banach spaces, we equip them with the floo norm: l i e J l l = max(ll e l l , I I J I I ) . B y a result of Loeb in [8] , there exists an internal *finite partition of *D consisting of sets A1 , . . . , AH E *I: (H E *N ) such that the partition is finer than the image under * of any finite partition of D into sets in I:. The proof uses a concurrent relation argument. From this fine partition we discard all partition sets of *!v i measure zero. The remaining sets still form an internal collection, which we also denote by A1 , . . . , AH . ::::J ,
1r
16.2
X
The exist ing literature
Ross gives a nice detailed nonstandard proof of the RadonNikodym the orem for two realvalued afinite measures fL < < v on a measurable space in [9] . Previously we have studied nonstandard RadonNikodym derivatives of vector measures tt : I: + E with respect to Lebesgue measure on [0, 1 ] . We have shown that through nonstandard analysis a unifying approach to
16.2. The existing literature
229 or
RadonNikodym derivatives independent of the RadonNikodym property lack thereof of E can be found. The generalized derivatives we constructed are not necessarily essentially separably valued and therefore not always Bochner integrable. However, a generalization of the Bochner integral in [10] for func tions with values in the nonstandard hull of a Banach space allows one to integrate the generalized derivatives. In [ 1 1] with methods similar to those used by Ross [9] or Bliedtner and Loeb in [3] , and with the use of local reflex ivity we "standardize" the generalized derivatives from maps *D + E to maps D + E" with values in the second dual of the original Banach space E. The vector measures book [5] by Diestel and Uhl is still considered as the authoritative work on RadonNikodym theorems. The standard literature yields very little on vectorvector derivatives; only Bogdan [4] together with his student Kritt has made an initial foray into this field. In their article they assume that they have two vector measures fL : I: + E and v : I: + F with fL < < I v 1 . and they assume that the range space of fL has the RadonNikodym property and the range space of v is uniformly convex. They then define a derivative and integral in the form fL(A) = fA ��I (w) t l (w) dv, where tl is a function from D into F', the dual space of the range of v . The integrand is then of the form �� ( w) = e . !' ' a function from n into E X F'. A trilinear product on E x F x F' with values in E can be defined as e · J' · f f' (f) · e. This is needed to integrate simple functions of the form L�=1 e, . !I · 1 A , : n + E X F' with respect to the Fvalued measure v as follows: ·
=
L t e; · JI · 1A, dv = t e; · JI · v(A; ) = t e, · J; (v(A; )) E E . ,=1
,=1
,=1
The main difficulty is the definition of tl , the derivative of a realvalued measure with respect to a vectorvalued measure, the opposite setting from the ordinary RadonNikodym theorems. Bogdan uses his assumption that F is uniformly convex to find this derivative. Since v < < l v l and since uniform convex spaces are reflexive and hence have the RadonNikodym property, there is a Bochner integrable function d��l : D + F such that for all measurable sets A, the vector measure can be written as a Bochner integral v(A) = fA d�� l dl v l . A standard theorem ( e.g. Theorem 11.2.4 in [5]) about vector measures states that the norm of the RadonNikodym derivative is a RadonNikodym derivative for the total variation of the vector measure, i.e. for all A E I: l v i (A)
=
i l d�� � � d l v l .
This implies that I vialmost everywhere on D, II d��1 1 1 1 . For uniformly convex spaces F there is a continuous map g : Sp + SF' , where Sx denotes =
230
16 .
A RadonNikodym theorem
the unit sphere of X, such that ( !, g(f)) = 1 for all f E Sp. In this case, the composition g o d�� l is a map from D into the unit sphere of F'. The derivative of fL with respect to v is defined as �� � · (g o d�� l ) : D + E F', or, if one regards the product e f' as a rank one continuous linear operator from F to E, the derivative can be considered a map: �� : D + L(F, E) . Since d��l and d�� l arc both Bochner integrable and g is continuous and g o d�� l is almost everywhere of norm one, it is easy to show that the result is integrable with respect to the vector measure v with an integral that uses approximation by simple functions. =
x
16.3
x
The nonstandard approach
With nonstandard analysis we can significantly weaken the assumptions on the Banach spaces E and F. We use Bogdan's idea of writing the derivative dlvl _ djV[ dp · ;IV dfl . (g o djV[ dv ) . as djV[ The derivatives ��I and d��l are derivatives of a vector measure with respect to a real valued measure. They can be found and integrated with the methods described in [10] , [11] or [12] . In [10] we defined a Banach space M(lvi, E) of e�ended integrable func tions as the set of equivalence classes under equality lvlalmost everywhere of functions f : *D + E for which there� an internal, *simple, Sintegrable f l vlalmost everywhere on *D. Such a cp : *D + *E such that 71£ o cp cp is called a lifting of f. On M ( lvl, E) we defined an integral by setting JA f d!vl 71 ( fA cp d * lvl ) , for all A E *I:, where the integral of the internal simple function cp is defined in the obvious way. This integral can be extended to sets in the Loeb aalgebra and generalizes the Bochner integral in the sense that M ( lvl, E) contains L1 ( lvl, E) and the integrals agree on that subspace. However, .iH(Ivi, E) also contains functions which fail to be essentially separa bly valued and hence fail to be measurable. Lemma 1 in [11] translates to this situation as: =
=
Let E be any Banach space and let fL : I: + E be a ! v i absolutely continuous countably additive vector measure of bounded variation. Then the internal *simple function cp 11 : *D + *E defined by
Lemma 16.3 . 1
is S integrable, where A1 , . . . , AH is the fine partition of *D introduced above.
16.3. The nonstandard approach
231
Sintegrability implies that the function is l v l almost everywhere fin(*E) valucd. This allows us compose an Sintcgrable internal function with the quotient map 71E : fin(*E) + E to make it a nonstandard hull valued function defined on the Loeb space ( *D , L i v i (*L: ) , !vi) . Define f11 E M(lvi, E) by
f!l = 71£ o CfJw
Theorem 4 in [11] asserts that if the vector measure fL has a Bochner integrable RadonNikodym derivative, then the generalized derivative "is" the Radon Nikodym derivative in the following sense:
Let fL : I: + E be a countably additive vector measure with a Bochner integrable RadonNikodym derivative f : D + E. Define the gener alized RadonNikodym derivative f11 : *D + E as above. Then 71£ o *f = f11 on each set Ai in the fine partition of *D.
Theorem 16.3.2
Of course, the same arguments also work for finding a Fvalued generalized derivative fv E lVI(Ivi, F ) of the vector measure v : I: + F with respect to its total variation I v i . Differentiating v with respect to its total variation gives one extra feature of the derivative:
Let F be a Banach space and let v : I: + F be a countably additive vector measure of bounded variation. Define
Lemma 16.3.3
Then fv = 71F o C{Jv is I vi almost everywhere of norm one. Since v 0 r =
J
Note that, if IL' and 1 ' are two internal derivatives of IL, then J g(w)d(IL')L (w) = J g( w )d(r')L for all g E C, but the Loeb measures ( �L ')L and (r')L on A and hence also the afields £.,, (A) and £1, (A) may be different. In Example 17.2.2 we defined an internal counting measure IL· Since IL is SSkorohoddifferentiable with internal derivatives T, t E J \ { 0 } , t � 0, we can17apply Lemma 17 17.3 . 1 . But as we have seen in Example 17.2.2, for k E *N with E J and � 0 and A = [0, k}/]
Example 1 7. 3 . 1
( IL Hk 17 IL ) (A) L
=
IL =!5:.  IL 0 and ( HI'l ) (A) L
=
1.
Nevertheless, there exists a afield, on which the Loeb measures ("1 �" ) L coin cide for all infinitesimals t E J \ { 0 } . Let B(ffi.) be the ( standard ) afield of all Borel subsets of R For B E B(ffi.) let sC 1 [B] = {w E D : 0W E B} . According to the usual approach to standard Lebesgue measure by a nonstandard count ing measure ( see [ 7] or [ 12] ) , the external set st  1 [B] is an element of L.,r (A) for all r E I and (ILr ) L ( s C 1 [B ] ) = vr ( B ) , where ur ( B ) = JB +r 1 [o . l j (x ) d>. (x ) (A) with standard Lebesgue measure >.. But st  1 [B ] is also an element of L for all infinitesimals t E J \ { 0 } and PrP t
( ILt � IL )
L
( sC1 [B ] ) = 1B( 1)  1 B (O) .
Hence the Loeb measures ( "1�" h , restricted to the afield {sC1 [B ] : B E B(ffi.)}, coincide for all infinitesimals t E J \ { 0 } . If we define a measure v' on B(ffi.) by v'(B ) = 1 B ( 1)  1B(O), then the Sdifferentiability of the internal counting measure tt yields the Skorohoddifferentiability ( with respect to 1 E ffi.) of the standard measure v with derivative v'.
17.3. Differentiability of Loeb measures
245
We will now see that in the case of SFomindifferentiability the afield :F doesn ' t depend on the chosen internal derivative and the derivative (ILL)' of ILL is uniquely determined. 1\Ioreover, the differentiability of ILL is true not only with respect to standard parts of internal functions, but also with respect to all :Fmeasurable Sbounded realvalued functions on D. We gather these facts together into the following, which is the main theorem of this paper.
Let 1L be SFomindifferentiable and :F = n rE I L11r (A). Then ILL is differentiable on :F with respect to the set C = { 1 B B E :F} and the differentiability is uniform on C. The derivative (ILL )' is uniquely de termined and is absolutely continuous with respect to ILL . If IL' is an internal derivative of IL, then the Loeb extension (IL')L is defined on :F and coincides with (ILL)' . In particular this is true for all internal measures � , where t E J \ { 0} is infinitesimal. Theorem 1 7. 3 . 1
:
Let IL' be a derivative of IL· We will show the following statements: (A) For all internal sets A E A the limit
Proof.
(ILL) ' ( A) ' �'LL (A) '  'll. m "r'r> 0 r exists and is equal to ( �L')L ( A) . The convergence is uniform on the inter nal field A. (B) If N E :F is a ILLnullset, then (ILL) r (N)  �LL (N) lim = 0. r r>0 The convergence is uniform for all ILLnullsets of :F. Any ILLnullset of :F is also a (IL')Lnullset of :F. (C) If B E :F and if A E A is ILLequivalent to B, then (ILL) r (A)  ILL ( A) (ILL) r (B)  ILL ( B) = lim lim ' r>0 r> 0 r r in particular the left limit exists. The convergence is uniform on :F. (D) The Loeb extension (IL')L is defined on :F and for all B E :F
(A) follows from Lemma 17.3.1. To prove (B) let N b e a ILLnullset of :F, i.e. there exists a sequence (N� )n EN C A so that for all n E N we have N C N1_ , Nn� c:;; N� and 1
1 7. Differentiability of Loeb measures
246
p,(Nl ) :
VT Vm
::J f z n c
[ [a (x) + a(v)
X
N c:;: I:
�
X
m(a, n)
e
V(x, v, t) 3 (p, n)
1\
0 < ltl
�
Ec
X
N
� ] =?
[T (df(x, v)) � n
1\
T( !:::. (h, x, v, t))
�
]
(18.4.6)
1] .
NB: variants of this formula where any inequality � is taken to be
strict, i.e., to be < , are equivalent.
18.
266
18.4.2
The power of Gateaux differentiability
The nonstandard hull
Theorem 18.4.2 Let E and F be locally convex spaces and f : *E + *F be
an internal GSdifferentiable function with derivative df such that f ( fin(*E))
( 18.4. 7)
� fin(*F) .
f is SUD iff j : E + F is uniformly differentiable with derivative dj : E2 + F given by d](x , v) := djr;;:; ) . Proof. Let us first assume that the internal function f : *E + *F is SUD with derivative df, and takes finite vectors of *E into finite vectors of *F. Define j3
Q := { I] I q E Q } ;
:= {P I p E P}
although there might exist continuous seminorms not of the type ,y, P and Q arc unbounded directed families of seminorms, so that condition ( 18. 1 .4) still holds with obvious adaptations. Also recall that
](x) = f(x) and observe that equation
defines dj well, in view of theorems 18. 1 . 5 and Take q E Q and rrL : Q x N + N . Define
M(p, n)
:=
m(p, n)
18. 1.6 and lemma 18.3. 1 .
(p E P, n E N) ;
M is a standard function. therefore we may deduce from exists a standard finite set P x N � *P x 'N such that
(18.4.5)
that there
V(x, v, t) E *E2 x 'R 3(p, n) E P x N
[p (x) + p(v) < M(p, n) 1\ 0 < l t l
: 0 be infinitesimal and a ao. Then every solution x(t) of the IVP (20.3.5) x(O) = a, x(t) f (tjr:: , x(t)) ' is a perturbation of y ( t), that is, for all nearstandard t in [0, w [, x( t) is defined and satisfies x(t) y(t). c:::
:
x
=
c:::
The proof, in the particular case of almost periodic vector fields, is given in Section 20.3.4. The proof in the general case is given in Section 20.3.5.
20.3.2
Almost solutions
The notion of almost solution of an ODE is related to the classical notion of c almost solution.
function x( t) is said to be an almost solution of the standard differential equation x G(t, x) on the standard interval [0, L] if there exists a finite sequence 0 to < < t N + 1 L such that for n 0, , N we have
Definition 4 A
=
=
c:::
·
·
·
tn+ l tn, x(t) and x(tn+ l )  x(tn) tn+ l  tn
=
=
·
c:::
x(tn) for t E [t n , t n+ l ] ,
c:::
G(tn , x(tn )).
·
·
The aim of the following result is to show that an almost solution of a standard ODE is infinitely close to a solution of the equation. This result which was first established by .J. L. Callot (sec [1 1 , 27]) is a direct consequence of the nonstandard proof of the existence of solutions of continuous ODEs [26] .
293
20.3. Averaging in ordinary differential equations
Theorem 3 If x ( t) is an almost solution of the standard differential equation x = G(t, x) on the standard interval [0, L] , x(O) y0, with y0 standard, and the IVP y = G(t, y), y(O) = y0, has a unique solution y(t), then y(t) is defined at least on [0, L] and we have x(t) y(t), for all t E [0, L] . "='
"='
D
Proof. See [ 1 1 , 36]
Let us apply this theorem to obtain an averaging result for an ODE which does not satisfy all the hypothesis of Theorem 2. Consider the ODE (see [ 1 1 , 27. 36] )
tx . . X. ( t ) = Sln c
(20.3.6)
The conditions (2) and (3) in Definition 2 are satisfied with F(x) = 0. Thus, the solutions of the averaged equation arc constant. But condition ( 1 ) of the definition is not satisfied, since the function f(t, x) = sin (tx) is not continuous in x uniformly with respect to t. Hence Theorem 2 docs not apply. In fact the solutions of (20.3.6) are not nearly constant and we have the following result: Proposition 2 If c > 0 is infinitesimal then, in the region t 2 x > 0 solutions of (20. 8. 6) are infinitely close to hyperbolas tx = constant . In region x > t 2 0, they are infinitely close to the solutions of the ODE
the the
2 2 (20.3. 7) x = G(t, x) , where G(t, x) = vlx  t  x . t Proof. The isocline curves h = { ( t, x) : tx = 2k7rc} and I� = { ( t, x) : tx = (2k + �) 7rc} define, in the region t 2 x > 0, tubes in which the trajectories are trapped. Thus for t 2 x > 0 the solutions are infinitely close to the hyperbolas tx = constant. This argument does not work for x > t 2 0. In this region, we consider the microscope
T
=
t  tk
X
=
c ' where ( tk , X k ) are the points where a solution h· Then we have
X  Xk c
x( t)
' of (20.3.6) crosses the curve
X(O) = 0. By the Short Shadow Lemma (Theorem 1 ) , X(T) is infinitely close to a solution of dXjdT = sin (x k T + t k X). By straightforward computations we have
X X tk + l  tk
k k+l '
":::
G(t k , X k ) .
Hence, in the region x > t 2 0, the function x( t) is an almost solution of the ODE (20.3. 7) . By Theorem 3, the solutions of (20.3.6) are infinitely close to the solutions of (20.3.7). D
20.
294
20.3.3
Averaging
The stroboscopic method for ODEs
�+
In this section we denote by G : x D + function, where D is a standard open subset of function such that 0 E I C Definition 5 We say that respect to G if there exists
�+.
�d.�d
�d
a standard continuous Let x : I + be a
x satisfies the Strong Stroboscopic Property with > 0 such that for every positive limited t0 E I with x(to) nearstandard in D, there exists t1 E I such that fL < t1  to 0, [to , t1] c I, x(t) x(to) for all t E [to, t1], and fL
c::::
c::::
x(t l )  x(to) G ( t0, x t0 )) . ( t1  to The real numbers t0 and t1 are called successive instants of observation of the �
'''' _
stroboscopic method. Theorem 4 (Stroboscopic Lemma for ODEs) Let a0 E D be standard. Assume that the IVP y(t) G (t, y(t)), y(O) a0, has a unique solution y defined on some standard interval [0, L] . Assume that x(O) a0 and x satisfies the Strong Stroboscopic Property with respect to G. Then x is defined at least on [0, L] and satisfies x(t) y(t) for all t E [0� L] . =
=
c::::
c::::
Proof. Since x satisfies the Strong Stroboscopic Property with respect to G, it is an almost solution of the ODE x = G(t, x). By Theorem 3 we have x(t) c:::: y(t) for all t E [0, L] . The details of the proof can be found in [36] . D The Stroboscopic Lemma has many applications in the perturbation theory of differential equations (see [11, 32, 35, 36� 38 , 39]). Let us use this lemma to obtain a proof of Theorem 2.
20.3.4
Proof of Theorem 2 for almost periodic vector fields
·,
Suppose that fo is an almost periodic in t, then any of its translates fo (s + x0) is a nearstandard function, and fo has an average F which satis fies [16, 28, 33, 41]
s+Tfo (t, x) dt, 1 T>rx> T s s E �+ · F 1 1 s+ T F(x) T s fo (t, x) dt,
F(x) uniformly with respect to
=
1
lim 
Since
c:::: 
is standard and continuous, we have
(20.3.8)
20.3.
295
Averaging in ordinary differential equations
IR
for all s E + , all T ::::::' oo and all nearstandard x in U0 . Let x : I + U be a solution of problem (20.3.5) . Let t0 be an instant of observation: t0 is limited in I, and x0 = x(to) is nearstandard in U0 . The change of variables
X transforms
(20.3.5)
=
_x'(t o + cTc) _ x_o ' c _ _ __
_
into
dX/dT
=
f(s + T, xo + eX) ,
s
where
=
tojc.
By the Short Shadow Lemma ( Theorem 1 ) , applied to g(T, X) f(s + T, xo + eX) and go (T, X) fo ( s + T, xo) , for all limited T > 0, we have X(T) ::::::' J0Y fo (s + r, xo)dr. By Robinson's Lemma this property is true for some unlimited T which can be chosen such that cT 0. Define t1 t0 + cT. =
=
::::::'
Then we have
=
1
l
+ y 1 s Y ::::::' 1 0 fo( s + r, xo) dr = T s fo (t, xo) dt ::::::' F(xo). T T Thus x satisfies the Strong Stroboscopic Property with respect to F. Using the Stroboscopic Lemma for ODEs ( Theorem 4) we conclude that x(t) is infinitely close to a solution of the averaged ODE (20.3.4). x(t l )  x(to) t1  to
'''' =
20.3.5
X ( T)

Proof of Theorem 2 for KBM vector fields
Let fo be a KBM vector field. From condition (2 ) of Definition 2 we deduce + that for all s E + , we have F(x) = limy_, = � fss Y fo (t, x) dt, but the limit is not uniform on s . Thus for unlimited positive s , the property ( 20.3.8 ) docs not hold for all unlimited T, as it was the case for almost periodic vector fields. However, using also the uniform continuity of fo in x with respect to t we can show that (20.3.8 ) holds for some unlimited T which are not very large. This result is stated in the following technical lemma [36] .
IR
IR
IRd
Let g : + x M + be a standard continuous function where M is a standard metric space. We assume that g is continuous in m E M uniformly with respect to t E + and that g has an average G(m) limy_, = � J0Y g(t, m) dt. Let c > 0 be infinitesimal. Let t E + be limited. Let m be nearstandard in M . Then there exists a > c, a 0 such that, for all limited T 2 0 we have + 1 s YS g(r, m) dr ::::::' T G ( m ) , where s = tjc, S = o:jc. Lemma 1
S
1 8
IR
::::::'
IR
=
20.
296
2
The proof of Theorem found also in [36] .
Averaging
needs another technical lemma whose proof can be
Let g IR+ JRd JRd and h IR+ JRd be continuous functions. Suppose that g(T, T X) h(T) holds for all limited T E IR+ and all limited X E JRd , and J0 h ( r) dr is limited for all limited T E IR+ . Then, any solution X(T) of the IVP dXjdTT g(T, X), X(O) 0, is defined for all limited T E IR+ and satisfies X(T) "=' J0 h(r) dr. Lemma 2
:
:
+
x
"='
=
+
=
Proof of Theorem 2. Let x : I + U be a solution of problem (20.3.5). Let t0 E I be limited, such that x0 = x(t0) is nearstandard in U0. By Lemma 1 , applied to g = fo, G = F and m = x(to), there is a > 0, a "=' 0 such that for all limited T 2 0 we have
T 1 rs + S fo (r, xo) dr "=' TF (xo), s .}s
where
s = to/c, S = o:jc.
(20.3.9)
The change of variables
X(T) transforms
(20.3.5)
=
x (to + o:T)  xo 0:
into
dXjdT
=
f(s + ST, xo + o:X).
By Lemma 2, applied t o g(T, X) = f(s and (20.3.9) , for all limited T >
+ ST xo + o:X) and h(T) = fo (s + 0, we have T s +TS X(T) "=' fo (s + Sr, xo) dr = 1 fo (r, xo) dr ":' TF (xo). 0 s s Define the successive instant of observation of the stroboscopic method t1 by t1 = to + a. Then we have ST, xo),
lo
1
x(t l )  x(to) = X ( 1) "=' F(xo). t1  to Since t1  to = a > c and x(t)  x(to) = o:X(T) "=' 0 for all t E [to, t1] , we have proved that the function x satisfies the Strong Stroboscopic Property with respect to F. By the Stroboscopic Lemma, for any nearstandard t E [0 , w[, x(t) i s defined and satisfies x(t) "=' y(t). D
20.4.
297
Functional differential equations
20.4
Functional differential equations
Let C = C ([  r, 0] , IRd ) , where r > 0, denote the Banach space of continuous functions with the norm 11 ¢ 11 = sup{ II ¢(B) II : e E [  r, O] } , where 1 1 · 11 is a norm of JRd . Let L 2 t0. If x : [r, L] + IRd is continuous, we define Xt E C by setting Xt (B) = x(t + B), e E [  r, OJ for each t E [0, L] . Let g : JR + x C + lRd , (t, u) f+ g(t, v. ) , be a continuous function. Let ¢ E C be an initial condition. A Functional Differential Equation (FDE) is an equation of the form
x(t) = g (t, xt) ,
xo = ¢.
This type of equation includes differential equations with delays of the form =
x(t) G(t, x(t), x(t  r) ) , where G : IR+ IRd IRd + JRd . Here we have g(t, u ) G(t, u(O) , u(  r)). The method of averaging was extended [13, 22] to the case of FDEs x
x
=
the form where
of
(20.4. 1)
c is a small parameter.
In that case the averaged equation is the ODE
(20.4.2) where F is the average of of the form
f.
It was also extended
x(t)
=
f (t/c, x t ) .
[14]
to the case of FDEs
(20.4. 3)
In that case the averaged equation is the FDE
(20.4.4) Notice that the change of variables x(t) = z(t/c) does not transform equa tion (20.4. 1) into equation (20.4.3) , as it was the case for ODEs (20.3.3) and (20.3.1), so that the results obtained for (20.4. 1) cannot be applied to (20.4.3) . In the case of FDEs of the form (20.4. 1) or (20.4.3), the clas sical averaging theorems require that the vector field f is Lipschitz continuous in x uniformly with respect to t. In our approach, this condition is weak ened and we only assume that the vector field f is continuous in x uniformly with respect to t. Also in the classical averaging theorems it is assumed that the solutions z(T, c) of (20.4. 1) and y(T) of (20.4.2) exist in the same interval [0, T /c] or that the solutions x(t, c) of (20.4.3) and y(t) of (20.4.4) exist in the same interval [0, T] . In our approach, we assume only that the solution of the averaged equation is defined on some interval and we give conditions on the vector field f so that, for c sufficiently smalL the solution x(t, c) of the system exists at least on the same intervaL
298
20. Averaging
20.4.1
Averaging for FDEs in the form z'(r)
=
E:f (r, Zr )
c is a small parameter zo c/J . The change of variable x(t) z(t/c) transforms this equation in (20.4. 5) x(t) f (tjc, Xt ,c ) , x(t) c/J(t/c), t E [  cr, 0] , where Xt , E E C is defined by Xt ,E (B) x( t + cB) for B E [  r, 0] . Let f IR+ x C + JRd be a standard continuous function. We assume that (H1) The function f : u f (t, u) is continuous in u uniformly with respect to the variable t. (H2) For all u E C the limit F(u) limrHx> � J0T f (t, u) dt exist s. We identify IRd to the subset of constant functions in C , and for any vector c E JRd , we denote by the same letter, the constant function u E C defined by u(B) c, e E [  r, 0] . Averaging consists in approximating the solutions x(t, c) of (20.4. 5) by the solution y(t) of the averaged ODE (20.4. 6) y(t) F (y(t)) , y(O) ¢(0) . According to our convention , y(t) , in the righthand side of this equation. is the constant function u t E C defined by ut (e) y(t), e E [r, O] . Since F is We consider the IVP, where
=
=
=
=
=
:
f+
=
=
=
=
=
continuous, this equation is well defined. We assume that
( H 3)
The averaged ODE (20.4.6) has the uniqueness of the solution with pre scribed initial condition.
(H4) The function f is quasibounded in the variable u uniformly with respect to the variable t, that is, for every t E IR+ and every limited u E C, f (t, u) is limited in JRd . Notice that conditions (H1), (H2) and ( H 3) are similar to conditions (1 ) , (2) and (3) of Definition 2 . In the case of FDEs we need also condition (H4) . In classical words, the uniform quasi boundedness means that for every bounded subset B of C, f (IR+ x B) is a bounded subset of IRd . This property is strongly related to the continuation properties of the solutions of FDEs (see Sections 2 . 3 and 3. 1 o f [1 5] ) . Theorem 5 Let f : IR+ x C + IRd be a standard continuous function satisfying
the conditions (Hl )  (H4). Let cjJ be standard in C. Let L > 0 be standard and let y : [0, L] + IRd be the solution of (20.4. 6). Let c > 0 be infinitesimal. Then every solution x(t) of the problem {20. 4 . 5) is defined at least on [ cr. L] and satisfies x(t) "::: y(t) for all t E [0 , L] .
20.4.
299
Functional differential equations
20.4. 2
The stroboscopic method for ODEs revisited
In this section we give another formulation of the stroboscopic method for ODEs which is well adapted to the proof of Theorem 5. Moreover , this formulation of the Stroboscopic Method will be easily extended to FDEs ( see Section 20.4.4) . We denote by G : IR x IRd + IRd , a standard continuous function. Let x : I + JRd be a function such that 0 E I C IR .
+
+
We say that x satisfies the Stroboscopic Property with respect to G if there exists fL > 0 such that for every positive limited t0 E I, satisfying [0, t0] C I and x(t) is limited for all t E [0, t0] , there exists t1 E I such that fL < t1  to "::: 0, [to, t1] c I, x(t) "::: x(to) for all t E [to , t1], and
Definition 6
x(t l )  x(to) t l  to
'''.'
_
�
G ( t0, x ( to )) .
The difference with the Strong Stroboscopic Property with respect to G con sidered in Section 20.3.3 is that now we assume that the successive instant of observation t1 exists only for those values t0 for which x(t) is limited for all t E [0, t0] . In Definition 5, in which we take D = IRd , we assumed the stronger hypothesis that t1 exists for all limited to for which x(to) is limited.
Let ao E D be standaTd. Assume that the IVP y(t) G (t, y(t)), y(O) ao, has a unique solution y defined on some standard interval [0, L] . Assume that x(O) "::: ao and x satisfies the Stroboscopic Property with respect to G. Then x is defined at least on [0, L] and satisfies x(t) y(t) for all t E [0, L] .
Theorem 6 (Second Stroboscopic Lemma for ODEs) =
=
c:::
Proof. Since x satisfies the Stroboscopic Property with respect to G, it is an almost solution of the ODE x = G(t, x). By Theorem 3 we have x(t) "::: y(t) for all t E [0, L] . The details of the proof can found in [19] or [21] . D Proof of Theorem 5. Let x : I + JRd be a solution of problem (20.4.5). Let t0 E I be limited, such that x(t) is limited for all t E [0, t0] . By Lemma applied to g = f, G = F and the constant function m = x(to), there is a > a "::: 0 such that for all limited T 2: 0 we have
1 S
ls+TS f(r, x(to)) dr "::: TF(x(to)), s
where
s = to/c, S = o:jc.
(20.4. 7)
Using the uniform quasi boundedness of f we can show ( for the details see or [21]) that x(t) is defined and limited for all t "::: t0 . Hence the function e
X ( , T)
=
x(to + aT + cB)  x(to) , B E [r, O] . T E [0, 1] , 0:
1, 0,
[19]
20.
300
is well defined. In the variable
X (  , T)
(20.4.5)
system
Averaging
becomes
ax (0, T) = f(s + ST, x(to) + aX(  , T)). oT Using assumptions (H1 ) and (H4) together with (20.4. 7), we obtain after some computations that for all T E [0, 1] , we have
X (O, T)
"'='
{T f(s + Sr, x(to)) dr = 1 1s+TSf (r, x(t0)) dr
Jo
S
8
"'='
TF(x(t0)).
Define the successive instant of observation of the stroboscopic method t1 = to + a. Then we have
t1
by
x(t i )  x(to) X (O, 1 ) F(x(to)). t l  to Since t1  to a > c and x(t)  x(to) aX(O, T) 0 for all t E [to, t1 ] , we have proved that the function x satisfies the Stroboscopic Property with respect to F. By the Second Stroboscopic Lemma for ODEs, for any t E [0, L] , D x(t) is defined and satisfies x(t) y(t). "'='
=
=
"'='
=
"'='
20.4.3
Averaging for FDEs in the form x (t)
We consider the IVP, where
x(t)
=
c
=
is a small parameter
f (tj c , Xt ) ,
xo
=
f (tjE:, Xt )
(20.4.8 )
¢,
We assume that f satisfies conditions ( H1 ) , ( H2 ) and ( H4 ) of Section Now, the averaged equation is not the O D E (20.4.6), but the FDE
y(t) = F ( Yt ) ,
20.4. 1.
(20.4.9)
Yo = ¢.
Averaging consists in approximating the solutions x(t, c ) of ( 20.4.8 ) by the solution y(t) of the averaged FDE (20.4.9). Condition (H3) in Section 20. 4. 1 must be restated as follows
(H3)
The averaged FDE ( 20.4.9 ) has the uniqueness of the solution with pre scribed initial condition.
d Theorem 7 Let f : IR+ C + IR be a standard continuous function satisfying the conditionsd (Hl)(H4) . Let ¢ be standard in C. Let L 0 be standard and let 0 be infinitesimal. y : [0, L] IR be the solution of problem (20.4.9). Let x
+
>
c >
Then every solution x(t) of the problem {20.4.8) is defined at least on [  r, L] and satisfies x(t) y(t) for all t E [r, L] . "'='
20.4.
301
Functional differential equations
2 0.4.4
The stroboscopic method for FDEs
Since the averaged equation (20.4. 9) is an FDE, we need an extension of the stroboscopic method for ODEs given in Section 20.4.2. In this section we denote by G : IR x C + IRd , a standard continuous function. Let x : I + IRd be a function such that [  r, OJ C I C IR .
+
+
We say that x satisfies the Stroboscopic Property with respect to G if there exists fJ· > 0 S1Lch that for every positive limited to E I, satisfying [0, to] C I and x(t) and G(t , Xt) are limited for all t E [0 , to] , there exists t 1 E I such that fJ < t1  to 0, [to, t1] C I, x(t) x(to) for all t E [to, t1] , and
Definition 7
c:::
c:::
x(t x(to) ' l )'  ' '  G ( to, X to ) . t1  to �
Notice that now we assume that the successive instant of observation t1 exists for those values t0 for which both x ( t) and G ( t, xt ) are limited for all t E [ 0, t0 ] . In the limit case r = 0 , the Banach space C is identified with IRd and the function X t is identified with x(t) so that , G(t , X t ) is limited, for all limited x(t) . Hence the "Stroboscopic Property with respect to G " considered in the previous definition is a natural extension to FDEs of the "Stroboscopic Property with respect to G " considered in Definition 6.
Let ¢ E C be standard. A s sume that the IVP y(t) G (t, Yt ), Yo ¢, has a unique solution y defined on some standard interval [ r, L] . Assume that the function x satisfies the Stroboscopic Property with respect to G and x0 c::: ¢. Then x is defined at least on [r, L] and satisfies x(t) y(t) for all t E [r, L] .
Theorem 8 (Stroboscopic Lemma for FDEs) =
=
c:::
Proof. Since x satisfies the Stroboscopic Property with respect to G, it is an almost solution of the FDE x = G(t, Xt ) · For FDEs, we have to our disposal an analog of Theorem 3. Thus x ( t) c::: y ( t) for all t E [0, L] . The details of the proof can found in [19] or [21] . D Proof of Theorem 7. Let x : I + JRd be a solution of problem (20.4.8) . Let t0 E I be limited, such that both x(t) and F(xt ) are limited for all t E [0, t0] . From the uniform quasi boundedness of f we deduce that x( t) is Scontinuous on [0, t0] . Thus Xt is nearstandard for all t E [0, t0] . By Lemma 1 , applied to g f, G F and m Xt 0 , there is o: > 0, o: c::: 0 such that for all limited T 2 0 we have =
=
1
S
1s +TS f (r, X 8
=
t 0 ) dr
c:::
T F ( Xt0 ) ,
where
s
=
to/c , S
=
o:jc.
(20.4.10)
20.
302
Averaging
Using the uniform quasi boundedness of f we can show ( for the details see or [2 1]) that x(t) is defined and limited for all t , t0. Hence the function
[19]
X ( e , T ) = x(to + aT + B)  x(to + B) , B E [r, 0] , T E [0, 1] , a is well defined. In the variable X ( · , T) system (20.4.8) becomes ax · oT (0, T) = f(s + ST, X t0 + aX( , T) ). Using assumptions ( H 1) and (H4) together with (20.4.10), we obtain that for all T E [0, 1] , we have
{T
1 X(O, T) , J f (s + Sr, xt 0 ) dr = s o
1s+ TS f (r, x ) dr , TF (xt ) . 0 t0 s
Define the successive instant of observation of the stroboscopic method t1 = to + a. Then we have
t1
by
x(t l )  x(to) X (O, 1) "' F ) (x to t1  to Since t1  to a > c and x(t)  x(to) aX (O, T) 0 for all t E [to, h ] , we have proved that the function x satisfies the Stroboscopic Property with respect to F. By the Stroboscopic Lemma for FDEs, for any t E [0, L] , x ( t ) is defined and satisfies x(t) , y(t) . D =
=
=
,
References
[1]
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[2]
H . BAR REAU and .J . HARTHONG (editeurs ) , dard, Editions du CNRS, Paris, 1989.
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Bifurcations,
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1990,
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[10] M . DIENER and G. WALLET (editeurs) , Mathematiques finitaires et anal yse non standard, Publication mathematique de l'Universite de Paris 7, Vol. 311, 312, 1989. [11]
J. L. CALLOT and T. SARI , Stroboscopie et moyennisation dans les sys tcmes d'cquations diffcrcntielles a solutions rapidement oscillantes, in
Mathematical Tools and Models for Control, Systems Analysis and Sig nal Processing, vol. 3, CNRS Paris, 1983.
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A . FRUCHARD and A. TROESCH (editeurs), Colloque Trajectorien a la memoire de G. Reeb et J. L. Callot, StrasbourgObernai, 1216 juin 1995, Prepublication de l'IRtvi A, Strasbourg, 1995.
[13]
J . K . HALE, "Averaging methods for differential equations with retarded arguments and a small parameter", .J. Differential Equations, 2 (1966)
57 73.
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ferential Equations, New York, 1993.
[16]
L . D . KLUGLER, Nonstandard Analysis of almost periodic functions, in Applications of Model Theory to Algebra, Analysis and Probabzhty, W.A.J. Luxemburg ed. , Holt, Rinehart and Winston, 1969.
[17]
M. LAKRIB, "The method of averaging and functional differential equa tions with delay", Int. J . Math. Sci. , 26 (2001) 497511.
[18]
M . LAKRIB, " On the averaging method for differential equations with delay", Electron. J. Differential Equations, 65 (2002) 116.
304
20. Averaging
Stroboscopie et moyenmsation dans les equations differen tielles fonctionnelles a retard, These de Doctorat en Mathematiques de
[19] 1\ I . LAKRIB,
l' Universite de Haute Alsace, Mulhouse, 2004. [20] 1\ I . LAKRIB and T. SARI, "Averaging results for functional differential equations", Sibirsk. Mat. Zh. , 45 ( 2004) 375386; translation in Siberian Math. J . , 45 (2004) 3 1 1320. [2 1] 1\ I . LAKRIB and T. SARI , Averaging Theorems for Ordinary Differential Equations and Retarded Functional Differential Equations. http : //www . math . uha . fr/ps/20050 1 lakrib . pdf [22] B . LEHMAN and S . P . WEIBEL . "Fundamental theorems of averaging for functional differential equations", J . Differential Equations, 152 ( 1 999) 160190. [23] C. LOBRY, 1 989.
Et pourtant . . . ils ne remplissent pas N !,
Aleas Editeur, Lyon,
[24] C. LOBRY, T. SARI and S. TOUHAMI, " On Tykhonov's theorem for con vergence of solutions of slow and fast systems", Electron. J. Differential Equations, 19 ( 1998) 122. [25] R. LUTZ and M . GOZE, Nonstandard Analysis: a practical guide applications, Lectures Notes in Math. 881, SpringerVerlag, 1 982.
with
[26] E . NELSON, "Internal Set Theory", Bull. Amer. 1\Iath. Soc . , 83 ( 1 977) 1 165 1 1 98. [27] G. REEB, Equations differentielles et analyse non classique ( d'apres .J . L. Callot) , in Pmceedings of the 4th International Colloquium on Differ ential Geometry ( 1 978) , Publicaciones de la Universidad de Santiago de Compostela, 1 979. [28] A. RoBINSON , " Compactification of Groups and Rings and Nonstandard Analysis". Journ. Symbolic Logic, 34 ( 1 969) 576588. [29] A. ROBINSON. 1 974.
Nonstandard Analysis,
American Elsevier, New York,
[30] .J .l\1. SALANSKIS and H . SINACEUR (eds.) , Le Colloque de Cerisy, SpringerVerlag, Paris, 1992.
Labyrinthe du Continu,
[31] J . A . SANDERS and F. VERHULST, Averaging Methods in Nonlinear Dy namical Systems, Applied Mathematical Sciences 59, SpringerVerlag, New York, 1985.
305
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T. SARI, "Sur la theorie asymptotique des oscillations non stationnaires", Asterisque 1091 10 (1983) 1411 58.
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T. SARI , Fonctions presque periodiques, in Actes de l 'ecole d 'ete Analyse non standard et representation du reel. OranLes Andalouses 1984. OPU Alger  CNRS Paris, 1985.
[34]
T. SARI, General Topology, in Nonstandard Analysis in Practice, F. Di ener and M. Diener (Eds. ) , Universitext, SpringerVerlag, 1995.
[35]
T. SARI, Petite histoire de la stroboscopie, in Colloque Trajectorien a la Memoire de J. L. Callot et G. Reeb, StrasbourgObernai 1995. Publication IRMA, Univ. Strasbourg (1995) , 515.
[36]
T . SARI, Stroboscopy and Averaging, in Colloque Trajectorien a la Me moire de J. L. Callot et G. Reeb, StrasbourgObernai 1995, Publication IRMA, Univ. Strasbourg, 1995.
[37]
T. SARI, Nonstandard Perturbation Theory of Differential Equations, Ed inburgh, invited talk in International Congres in Nonstandard Analysis and its Applications, ICMS, Edinburgh, 1996. http : //www . math . uha . fr/sari/papers/i cms 1996 . pdf
[38]
T . SARI , Averaging in Hamiltonian systems with slowly varying parame ters, in Developments in Mathematzcal and Experimental Physics, Vol. C, Hydrodynamics and Dynamical Systems, Proceedings of the First Mexican Meeting on Mathematical and Experimental Physics, El Colegio Nacional, Mexico City, September 1014, 2001 , Ed. A. Macias, F. Uribe and E. Diaz, Kluwer Academic/Plenum Publishers, 2003.
[39]
T . SARI and K . YADI, "On PontryaginRodygin's theorem for convergence of solutions of slow and fast systems", Electron. J. Differential Equations, 139 (2004) 117.
[40]
I. STEWART,
[41]
K . D . STROYAN and W . A . .J . LUXEMBURG , infinitesimals, Academic Press, 1976.
[42]
I I I8
1987.
The Problems of Mathematics,
Oxford University Press,
Introduction to the theory of
rencontre de Geometrie du Schnepfenried, Feuilletages, Geometrie symplectzque et de contact, Analyse non standard et applications, Vol. 2, 1015 mai 1982 , Asterisque 109110, Societe Mathematique de France, 1983.
Pathspace measure for stochastic differential equation with a coefficient of polynomial growth Toru Nakamura
A a additive measure over
*
Abstract
a spa.ce of paths
is constructed to give the so lution to the FokkerPlanck equation associated with a st.oehastic differ ential equation with coefficient fuuction of polynomial growth by making use of nonstauda.rd analysis.
21.1
Heuristic arguments and definitions
Consider a stochastic differential equation,
dx(t)
=
f (x(t)) dt + db(t),
(21 . 1 .1 )
where f ( x) is a realvalued function and b(t) a Browni an motion vvith vari an ce 2Dt for a time interval t. vVe wish to construct a m easme over a space of paths for (21 . 1 .1 ) . To my knowledge, the coefficient. fund.ion has been assumed to have at most linear growth, that is [.f(x) [ .::::; const· [ x [ for sufficiently lar ge x, otherwise some paths explode to infinity in finite tunes. As an example, let .f(x) = [ x [ l+6 (8 > 0) and define explosion t ime e for each continuous path x(t) by limt>eo x(t) = ±oo. Then, it is proved that P(e oo) < 1 and more strongly P( e = oo ) 0. For general case, see Feller's test for explosion in [1]. Despite the explosion, we shall consider f(x) of polynomial growth of an arbitrary order and define a measure over a space of paths. vVe use nonstandard analysis because it has a very convenient theory, Loeb measure theory [2, 3] , which enables us t.o construct. a standard O"additive measure in a simple way. ·
=
=
*Department of Mat.hemat.ics. S undai Preparatory School, KandaS urugadai. Chiyoda kn, Tokyo 1 0 10062, .Japan.
21.1.
307
Heuristic arguments and definitions
Let us interpret (21 . 1 . 1) as a law for a particle momentum x = x(t) at time t with the force f ( x( t)) for drift and the random force db( t) / dt acting on the particle. By a time t > 0 some particles may disappear to infinity along the exploding paths, but others still exist with finite momentum so that they should make up a "probability" density of a particle momentum. We wish to usc the word "probability" though its total value may be less than 1 because of the disappearance of particles to infinity. Especially when f ( x( t)) is a repulsive force, that is, its sign is the same as that of x(t) , a particle can get larger momentum compared to the case where f (x(t)) is absent. However, once it gets very large momentum, it can hardly come back to the former one because the drift force f (x(t)) acts on the particle so as to increase its momentum. Thus, the particles which have gone to infinity by a time t > 0 could not contribute to the probability density at t. This consideration suggests that we can introduce a cutoff at an infinite number in momentum space if it is necessary in order to define a measure over a space of paths so that the probability density should be constructed by a path integral with respect to the measure. Eq. (21.1.1) gives the forward FokkerPlanck equation for the probability density U(t, x) of a particle momentum x at time t, 0
0 2 D o 2 U(t, x)  { J(x) U(t, x) } . (21 . 1 .2) ot ox ox We assume that the drift coefficient f(x) and the initial function U(O. x) satisfy
U(t. x)
=
the following conditions:
(A1)
For some natural number large x.
n
E N,
l f(x) l
:
.
.
22.
348
[12]
Optimal control for NavierStokes equations
N . J . CuTLAND and H . J . KEISLER, "At tractors and neoattractors for 3D stochastic NavierStokes equations", Stochastics and Dynamics, 5 (2005)
487533.
[13] [14]
G . DAP RATO and A. DEBUSSCHE. " Dynamic programming for the stochastic N avierStokes equations", Mathematical Modelling and N umer ical Analysis, 34 (2000) 459 475. G . DAPRATO and J . ZABCZYK , Stochastic Equations Cambridge University Press, Cambridge, 1992.
sions,
in Infinite Dimen
[15]
R.V. GAMKRELIDZE,
[16]
K. G RZESIAK, Optimal Control for NavierStokes Equations using Non standard Analysis, PhD Thesis, University of Hull, UK, 2003.
[17]
A . ICHIKAWA , " Stability of semilinear stochastic evolution equations", Journal of f\Iathcmatical Analysis and Applications, 90 (1982) 1244.
[18]
S . S . SRITHARAN (ed. ) , Optimal Control of Viscous in Applied Mathematics, Philadelphia, 1998.
[19]
R. TEMAM, NavierStokes edition, 1984.
[20]
M. VALADIER, Young measures, in Methods of Nonconvex Analysis (A. Cellina, ed. ) , Lecture Notes in f\Iathematics 1446, SpringerVerlag, Berlin,
1978.
1990.
Principles of Optimal Control, Plenum, New York,
Equations,
Flow, SIAM Frontiers
NorthHolland, Amsterdam, third
Lo calintime existence of strong solutions of the ndimensional Burgers equation via discretizations Joao Paulo Teixeira*
Consider the u1.
=
Abstract
e quation :
111l11.
 (·u V')u + f ·
for
a:
E
[0, 1]"
and t E
(01
)
x ,
together v.rith periodic boundary conditions and initial condition ·u ( t.. 0)
_q(:c) .
=
This corresponds a NavierStoket; problem where t he incompress ibility condition has been dropp ed . The maj or difficulty iu existence proofs for this simplified problem is the unbo unded advection term,
(n · \7)'!1..
We present a proof of localintime existence of a smooth solution based on a di scr e ti zation by a suitable Euler scheme. It will be shown that this solution exists in an int erval with C depending only on n and tlte values of the Lipschitz constants of f and u a.t time The argument given is based directly on local estimates of the solutions of the discrctizcd problem.
[0. T) , where T ::; t;,
23 . 1
0.
Introduction
The Burgers equation 'Ut
 II.I::!..U + (u \l)·u = .{ ·
x E D c �" , t > O
provides an example of a model for flows that takes into account the interaction between diffusion and ( nonlinear ) advection. This is probably the simplest nonlinear physical model for turbulence. *
lllstitnto Superior Tecn.ico, Lisbon, Portugal . j teix@math . ist . utl . pt
350
23. Burgers equation via discretizations
This equation is a simplification of the NavierStokes equations where the incompressibility constraint on the flow has been dropped. The NavierStokes equations are usually studied by taking projected solutions of the Burgers equations onto a subspace of divergence free functions. The main difficulty in the analysis of nonlinear flows, the advection term, ( u · 'V)u, is present in both equations. It is a consequence of the incompressibility condition that there are only trivial NavicrStokes flows for n = 1 . This is not the case in Burgers flows, where the case n = 1 is already nontrivial. Cole [6] and Hopf [10] have studied the problem in the real line:
{
Ut  VUxx + UUx u(x, t) = uo (x)
=
0
X
E
X
ER
IR, t > 0
(23. 1 . 1 )
Using the transformation of variables
v u = 2v ,
(23 . 1 .2)
V.r
(23. 1 . 1) was reduced to and initial value problem for the heat equation:
{
Vt  VVxx = 0
v(x, t)
=
vo(x)
X
E
X
ER
IR, t > 0
(23 . 1 .3)
(Where u� = 2 v ..!jvo ) . This enabled them to get a formula for a solution of problem (23. 1 . 1 ) . in terms of a Gaussian integral. Thus, and under very mild conditions on u0 , a solution, u, exists for all x E lR and t > 0, and is smooth. It would be reasonable to expect that the n > 1 case should behave in a similar way. The usual standard theory for this type of equations uses a weak formulation of the problem in Sobolev spaces, a Galerkin approximation to show existence of weak solutions and, finally, regularity estimates. How ever, these methods only lead to partial results. For wellposed problems with generic initial and boundary data, the best that can be obtained is local in time existence of regular solutions. It can also be shown that regular (strong) solutions are unique (if they exist) . See [4, 13, 9, 12, 16] for accounts on these methods. To gain some new insight into this problem, we develop a hyperfinite ap proach to the following model problem on a compact domain.
Let 'll' n = IRn ;zn be an ndimensional torus. Assume that f is locally Lipschitz continuous on 'll'n X [0, oo ) and u0 E C2 •1(1!'n ) (that is, uo is twice differentiable and all its second partial derivatives are Lipschitz continuous on 'll'n ). Let D = 'll'n X (0, oo ) . Let E JR+ . Let f IRn ;zn + IR, Model Problem:
v
:
351
23. 1 . Diffusionadvection equations in the torus
u0 : JRn ;zn + JRn . Our task is to study the initial value problem for the Burgers equations:
{
Ut  vb.. u + (u · 'V)u = f u = uo
D on 'll'n X {0}. in
(23. 1 . 4)
A (strong) solution of this problem is a sufficiently smooth u satisfying (23 . 1 . 4) . By "sufficiently smooth u" we mean that u : 'll'n x [0, oo ) + lR is such that u(  , t) E C2 ( 1l'n ) for all t E [0, oo ) and u ( x , ·) E C 1 ( [0. oo ) ) for all x E 'll'n .
23 . 2
A discretization for the diffusionadvection equations in the torus
We now look for a discretized version of problem (23 . 1 .4) . Since we are mainly interested in the existence result, we will do this in the simplest way possible. First we work in the standard universe. To discretize 'll'n , we introduce an hspaced grid on 'll'n . Choose lvf E N 1 , and let h = Jv1 . Then, let: 11':\1
=
{ 0, h, 2h, . . . , (M  l ) h , 1 r
=
h (Z mod Mt.
Consistently with our interpretation of 'll'M as a discrete version of the toms, given any x ( m 1 , m2 , . . . , mn ) h and
'll'n , we define addition in 1!'�1 as follows: y = (h, l2 , . . . , ln )h in 11':\1 , let:
=
This makes addition welldefined in 'll'M ; furthermore, it will behave in a similar way to addition in 'll'n . In particular, the set of gridneighbors of any x E 'll'M ,
{x ± hei :
i = 1, 2, . . . , n }
is welldefined. As for the discretization of time, consider T and define:
lf:
=
{ 0, k, 2k, . . . , (K  l)k, T }
E JR+ and K E N 1 . =
k (N n [0, K) );
Let
k
=
J? ,
To each triple d = (M, K, T), with M , N E N 1 and T E JR+ , we associate a discretization as defined above . We will , later on, introduce some restrictions on the set of admissible d. For now, given any d = (lvf, K, T) , with M, N E N 1 and T E JR+ , we let :
352
23. Burgers equation via discretizations Dd
=
']['M
X
(II u {T})
The elements of Dd are called gridpoints. Any function U whose domain is a subset of Dd is called a gridfunction. The discrete Laplacian can be defined as a map b.. d : JR15d + JR15d given by:
(
)
1 n U(x + h e,, t)  2U(x, t) + U(x  h ei , t) . (23.2. 1 ) h2 L i=l The definition of addition in 'll'M makes this welldefined. The discrete version of the nonlinear parabolic operator, P, that occurs in the diffusionadvection equations is defined as the map Pd : JRDd + JRDd b.. d U(x, t)
=
given by:
" U(x, t + k)  U(x, t)  vudU ( x, t ) k U(x + he , , t)  U(x  he , , t) . L....., U · ( x, t ) +� 2h i= l '
With we get:
PdU(x, t) �
=
,�;,, (U(x, t + k)  ( 1  2nvA)U(x , t) 
 >.
� ((
( �
)
�
(23.2.2)
)
))
v  U; (x, t) U(x I he ; , t) + v + U; (x, t) U(x  he ; , t)
The discretized version of problem (23 . 1 .4) is, then:
{
PdU(x, t) f(x, t) U(x, 0) uo(x, 0) =
=
if
(x, t)
if
X E
E
Dd
1l'A1 ·
(23.2.3)
Let us look more closely at the finite difference equation in (23.2.3). If we solve for U(x, t + k), we get: =
U(x, t + k) (1  2nv.A) U(x, t) n (, h h t) U(x+hei , t) + v+ 2 Ui (x, t) U(x  he i . t)�) U (x, +A v i 2 \ + .Ah 2 f(x, t)
�(
)
(
)
(23.2.4)
23.2.
353
Standard estimates for the discrete problem
Equation (23.2.4), together with the initial condition in (23.2.3), gives a recursive formula for the unique solution of problem (23.2.3). Define a map
by:
+ + � (0  � v, (x t)) U(x+he, t) + 0+�V,(x, t)) U(x  he,, ,v,
(U, V)(x, t)
=
(1  2nvA) U(x, t)
.x
,
whenever
(x, t)
E Dd . Equation
+
U ( x, t k)
=
(23.2.4)
(23.2.5)
can now be written as:
+
. =
=
v
(1  2n .A ) v(x , t) +
� ((v � a, (x, t) ) v(x + he; , t) + (v + � a; (x, t) ) v(x  he, , t)) +
+ .Ah 2 f(x , t) + r:: .A h 2 m for all natural m = 1 , 2, 3. · · · =


=
Figure 24.2.3: 63 on a 62 scale Laws of algebra dictate many "orders of infinitesimal" such as 0 < · · · < 63 < 62 < 6. The laws of algebra also show that near every real number there are many hyperreals, say near 3.14159 · · · 1r =
...
0 =? (r = U[�] , l r  xo l < � ' f[r] is undefined ) is true. The Function Extension Axiom means it must also be true with �
=
6 :::::J 0, a contradiction, hence, there is a positive real � so that f[x] is defined whenever lx  xo l < �.
We complete the proof of the Inverse Function Theorem by a permanence principle on the domain of yvalues where we can invert f[x] . The intuitive proof of Section 1 shows that whenever I Y  Yo I < 6 :::::J 0, we have lx 1  xo l :::::J 0, and for every natural n and k,
l xn  xo l < 2 1 x 1  xo l , l xn+ k  x k l < 2 k1_1 l x 1  xo l , f [xn ] is defined, I Y  f[xn+ l l l < � � Y  f[xnJ I Recall that we refocus our infinitesimal microscope after each step in the recursion. This is where the uniform condition is used. Now by the permanence principle, there is a real � > 0 so that whenever I Y  Yo I < � ' the properties above hold, making the sequence Xn convergent. Define g[y] = n+ Limrx> Xn · D 24. 2 . 12
Second differences and higher order smoothness
In Section 1 we derived Leibniz' second derivative formula for the radius of curvature of a curve. We actually used infinitesimal second differences, rather than second derivatives and a complete justification requires some more work.
References
393
One way to restate the Uniform First Derivative Theorem above is: The curve y = f[x] is smooth if and only if the line through any two pairs of infinitely close points on the curve is near the same real line, X l ::::::: X2 =}
f[x l ]  f[x 2 ]
::::::: m
X l  X2
A natural way to extend this is to ask: What is the parabola through three infinitely close points? Is the (standard part) of it independent of the choice of the triple? In [8] , Vftor Neves and I show: Theorem 12 Theorem on Higher Order Smoothness
Let f[x] be a real function defined on a real open interval (a, w). Then f[x] is ntimes continuously differentiable on (a, w) if and only if the nth _order differences 6n f are Scontinuous on (a , w) . In this case, the coefficients of the interpolating polynomial are near the coefficients of the Taylor polynomial, whenever the interpolating points satisfy x 1
:::::::
·
·
·
::::::: Xn
::::::: b.
References
[1]
MICHAEL BEHRENS , A Local Inverse Function Theorem, in Victoria Sym Lecture Notes in Math. 369, Springer Verlag, 1974.
[2]
H . J . M . Bos, "Differentials, HigherOrder Differentials and the Derivative in the Leibnizian Calculus", Archive for History of Exact Sciences, 14 1974.
[3]
L . EULER, Introductio in Analysin Infinitorum, Tomus Primus, Lausanne, 17 48. Reprinted as L. Euler, Opera Omnia, ser. 1, vol. 8. Translated from the Latin by .J. D. Blanton, Introduction to Analysis of the Infinite, Book I, SpringerVerlag, New York, 1988.
[4]
JON BARWISE (editor) , The Handbook of Mathematical Logic, North Hol land Studies in Logic 90, Amsterdam, 1977.
[5]
BERNARD R. GELBAUl\1 and JOHN M . H. OLMSTED , in Analysis, HoldenDay Inc. , San Francisco, 1964.
[6]
H . JEROME KEISLER, Elementary Calculus: An Infinitesimal edition, PWS Publishers, 1986. Now available free at http : //www . math . wisc . edu/ ke i sler/calc . html
posium on Nonstandard Analysis,
2nd
Counterexamples Approach,
24.
394
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Calculus with infinitesimals
l\ I ARK McKINZIE and CURTIS T C CKEY , "Higher Trigonometry, Hyper real Numbers and Euler's Analysis of Infinities". Math. Magazine, 74
( 2001) 339368.
[8]
VITOR NEVES and K. D . STROYAN, "A Discrete Condition for Higher Order Smoothness", Boletim da Sociedade Portugesa de Matematica, 3 5
(1996) 8194.
[9] ABRAHAM ROBINSON , "Nonstandard Analysis.. , Proceedings of the Royal Academy of Sciences, ser A, 64 ( 1961 ) 432440 [10] ABRAHAM
ROBINSON, Nonstandard Analysis, NorthHolland Publishing Co. , Amsterdam, 1966. Revised edition by Princeton University Press, Princeton, 1996.
[11]
K . D . STROYAN, Projects for Calculus: The Language of Change, on my website at http : //www . math . uiowa . edu/%7Estroyan/Proj e ct sCD/estroyan/ indexok . htm
[12]
K . D . STROYAN, Foundations of Infinitesimal Calculus. http : //www . math . uiowa . edu/%7Estroyan/backgndct lc . htm
[13]
K . D . STROYAN and W . A . .J . LUXEl\IBURG , Introduction to the Theory of Infinitesimals. Academic Press Series on Pure and Applied Math. 72, Academic Press, New York, 1976.
PreUniversity Analysis Richard 0 )Donovan *
This paper is
a
Abstract
Hrbacek's article showin g
how his
ap
Conceptual difficulties arise in elementary pedagogical approaches.
In
followup of
proach can be pedagogically tmiversity level.
K.
helpful
when introducing analysis at pre
most cases it remains difficult to explain at preuniversity level how the
derivative is calculated at nonstandard values or how
an
internal func
tion is clefiucd. 1Irbacek provides a modified version of 1ST lSI
(rather
P6raire's RIST) which seems to reduce all tlw,se difficulties. This system is hricHy
preR ented
here i11 its pedagogical form with
an
app licat io n to
the derivative. It must be understood as a stateoftheart report1 .
25.1
Introduction
Infinitesimals aTe interesting when teaching analysis because they give meaning to symbols s uch as dx, dy or df(x) and all related formulae. Also, symb ol ic manipulations are usually simpler thtn with limits. In secondary school, real numbers arc introduced with no formal justifica tion. It is sometimes sh ow n that v'2 tJ_ infinitesimal h , with x + h E ]a; b[, S
h < x ,j>
( f(x
+ h) h
f(x)
)
=
L
then the derivative is f' (x) = L It is a direct observation that the derivative is "acceptable". For f : x f+ x 2 at x = 2 the derivative is simple and works exactly as in any other nonstandard method. For x in general. the quotient simplifies to 2x + h. The only parameter of f is 2 E v(O) hence v(x, f) v(x). As h is xinfinitesinml, shx (2x + h) = 2x. For a direct calculation of f' (2 + c5), we have 2 + c) E v(c5) hence a [2 + 6]infinitesimal is also a 6infinitesimal. Let h be a 6infinitesimal.
=
Thus stratified analysis satisfies one of our major requirements: a single definition of the derivative which applies to all numbers. In [10] it is shown that using the definition of infinitesimals, the students could work out the rules of computation as exercises and find that for standard a and infinitesimal c), a · c) is infinitesimal. The same exercises adapted to ac ceptable statements yield that if 6 is ainfinitesimal, then a · 6 is ainfinitesimal.
25 . PreUniversity Analysis
400
25.5
Transfer and closure
Some definitions have been rewritten several times and the definitions we use today may not be final. All the proofs of theorems in the syllabus must be checked and crosschecked. The only principles that are needed are Hrbacek's closure principle: If f is an accept able function, then f ( x ) E v ( x, f) for all x in the domain of f. This extends the observation that operations with "familiar" numbers do not yield infinitesimals. The other is a simple form of transfer which states that an acceptable statement is true for v ( a ) iff it is true for all v ((J) with a E v ( (J ) . This adaptation to preuniversity level is not completed yet . We hope to finish a first version of a handout , with proofs, within a year or so. Our goal is still t he same: for most people, infinitesimals "are there", but can they be used to make maths easier to teach and learn and still remain rigorous? We hope to be able to contribute an answer to this question in a not too distant future.
References [1] Occam's razor, In
Encyclopaedia Britannica,
1 998. CD version.
[2] VIERI BENCI , MARC FORTI and MAURO Dr NASSO, The eightfoldpath to nonstandard analysis, Quaderni del Dipartimento di :t\1athematica Ap plicata "U. Dini", Pisa, 2004. [3] MARTIN BER Z, "Nonarchimedean analysis and rigorous computation", International .Journal of Applied Mathematics, 2 (2000) 889930. [4] MARTIN BERZ, " Cauchy theory on LeviCivita fields", Contemporary Mathematics, 319 (2003) 3952.
Analyse Non Standard, Hermann , 1989. Standard and Nonstandard Analysis, Ellis Horwood,
[5] F . DIENER and G . REEB , [6] RoY HOSKINS ,
1990.
[7] RoY HOSKINS, 2004. personal letter. [8] KAREL HRBACEK , Stratified analysis? , in this volume. [9] .JEROME KEISLER, Elementary Calculus, University of Wisconsin, 2000. www . math . wi s c . edu/ke isler/cal c . html [10] JOHN KIMBER and RICHARD O ' DONOVAN, Non standard analysis at preuniversity level, magnitude analysis. In D. A. Ross N . .J . Cutland, M . D i Nasso (eds.), Nonstandard Methods and Applications in Mathematics, Lecture Notes in Logic 25, Association for Symbolic Logic, 2006.
References
401
[ 1 1]
MAURO Dr NASSO and VIERI BENCI, Alpha theory, an elementary ax iomatics for nonstandard analysis, Quaderni del Dipartimento di Mathe matica Applicata "U. Dini", Pisa, 200 1 .
[ 1 2]
CAROL SCHUMACHER, Chapter Zero,
[13]
KHODR SHAMSEDDINE
[14]
K . D. STROYAN, Mathematical Background: Foundations of Infimtesimal Calculus, Academic Press, 1997.
AddisonWesley. 2000.
and MARTIN BERZ, "Intermediate values and inverse functions on nonarchimedean fields", IJMMS, (2002) 1651 76.
www . math . u iowa . edu/%7Estroyan/backgndctlc . html