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




we have almost everywhere

This implies our desired conclusion:

Problem 5[edit]

Solution 5[edit]