UMD Analysis Qualifying Exam/Jan09 Real
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Problem 1[edit | edit source]
(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[edit | edit source]
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: