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UMD Analysis Qualifying Exam/Jan09 Real
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Problem 1
(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
Problem 3
Let and suppose . Set for . Prove that for almost every ,
Solution 3
Change of variable
By change of variable (setting u=nx), we have
Monotone Convergence Theorem
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
Since
,
we have almost everywhere
This implies our desired conclusion:
Problem 5
Solution 5