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UMD Analysis Qualifying Exam/Jan09 Real

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Problem 1

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(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

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Problem 3

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


Solution 3

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Change of variable

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


Monotone Convergence Theorem

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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

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Since


,


we have almost everywhere



This implies our desired conclusion:


Problem 5

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Solution 5

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