UMD Analysis Qualifying Exam/Jan09 Real

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  

Conclusion edit



we have almost everywhere


This implies our desired conclusion:


Problem 5 edit

Solution 5 edit