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
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Solution 1
[edit | edit source]Problem 3
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Let and suppose . Set for . Prove that for almost every ,
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Solution 3
[edit | edit source]Change of variable
[edit | edit source]By change of variable (setting u=nx), we have
Monotone Convergence Theorem
[edit | edit source]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 | edit source]Since
,
we have almost everywhere
This implies our desired conclusion: