UMD Analysis Qualifying Exam/Jan09 Real

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Problem 1[edit]

(a) Let be real valued measurable functions on with the property that for every , is differentiable at and

Prove that


(b) Suppose in addition that is bounded on Prove that

Solution 1[edit]

Problem 3[edit]

Let and suppose . Set for . Prove that for almost every ,


Solution 3[edit]

Change of variable[edit]

By change of variable (setting u=nx), we have


Monotone Convergence Theorem[edit]

Define .


Then, is a nonnegative increasing function converging to .


Hence, by Monotone Convergence Theorem and



where the last inequality follows because the series converges ( ) and

Conclusion[edit]

Since


,


we have almost everywhere



This implies our desired conclusion:


Problem 5[edit]

Solution 5[edit]