UMD Analysis Qualifying Exam/Aug07 Real

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Problem 1[edit]

Suppose that is a continuous real-valued function with domain and that is absolutely continuous on every finite interval .


Prove: If and are both integrable on , then

Solution 1[edit]

Since is absolutely continuous for all ,



Hence



Since is integrable i.e. , and exist.


Assume for the sake of contradiction that



Then there exists such that for all



since is continuous. (At some point, will either monotonically increase or decrease to .) This implies



which contradicts the hypothesis that is integrable i.e. . Hence,



Using the same reasoning as above,



Hence,




Alternate Solution[edit]

Suppose (without loss of generality, ). Then for small positive , there exists some real such that for all we have . By the fundamental theorem of calculus, this gives

for all .


Since is integrable, this means that for any small positive , there exists an such that for all , we have . But by the above estimate,

This contradicts the integrability of . Therefore, we must have .

Problem 3[edit]

Suppose that is a sequence of real valued measurable functions defined on the interval and suppose that for almost every . Let and and suppose that for all


(a) Prove that .

(b)Prove that as


Solution 3a[edit]

By definition of norm,


Since ,



By Fatou's Lemma,



which implies, by taking the th root,



Solution 3b[edit]

By Holder's Inequality, for all that are measurable,



where


Hence,

The Vitali Convergence Theorem then implies


Problem 5[edit]

Suppose . Prove that and that


Solution 5[edit]