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

Since


 ,


we have almost everywhere


 


This implies our desired conclusion:


 

Problem 5

edit

Solution 5

edit