Real Analysis/Darboux Integral
|←Fundamental Theorem of Calculus||Real Analysis
Another popular definition of "integration" was provided by Jean Gaston Darboux and is often used in more advanced texts. In this chapter, we will define the Darboux integral, and demonstrate the equivalence of Riemann and Darboux integrals.
Upper and Lower Sums
Let be a (finite) partition of
For every define:
The Upper Sum of with respect to is defined as
The Lower Sum of with respect to is defined as
A partition is said to be a Refinement for a given partition iff
Let be a partition and let be a refinement of . Then,
Let and let be such that . Also, let , and
Obviously, , but as is arbitrary, we have that
Similarly, we can prove
We say that is Darboux Integrable on if and only if
, where the supremum is taken over the Set of all partitions on that interval
is also written as
is Darboux integrable over if and only if for every , there exists a partition on such that
()Let and let be given. Thus, by Gap Lemma, there exists a partition such that both , and hence
()Let be any partition on . Observe that is a lower bound of the set is any partition and that is an upper bound of the set is any partition
Thus, let and . As , we have that cannot be true. Also, as are a supremum and infimum respectively, is also not possible. Hence, (say).
As , we have that
Equivalence of Riemann and Darboux Integrals
At first sight, it may appear that the Darboux integral is a special case of the Riemann integral. However, this is illusionary, and indeed the two are equivalent.
(1) Let be Darboux Integrable, with integral
Let . Consider set of tagged partitions such that
Let be the set of where and
note that and that the set indeed contains all partitions with
Now, for , we can construct such that
(1) is Riemann integrable on iff
(2) is Darboux Integrable on
() Let be given.
(1) tagged partition such that .
Let partitions and be the same refinement of but with different tags.
i.e., by the triangle inequality,
Gap Lemma ,
being arbitrary, using Theorem 2.1, we have that is Darboux Integrable.
()Let be given.
(2), Theorem 2.1 partition such that
By Lemma 3.1, if
Thus, if we put , we have (1)
We note here that the crucial element in this proof is Lemma 3.1, as it essentially is giving an order relation between and , which is not directly present in either the Riemann or Darboux definition.