UMD Analysis Qualifying Exam/Aug07 Real

Problem 1 edit

Suppose that   is a continuous real-valued function with domain   and that   is absolutely continuous on every finite interval  .

Prove: If   and   are both integrable on  , then


Solution 1 edit

Since   is absolutely continuous for all  ,




Since   is integrable i.e.  ,   and   exist.

Assume for the sake of contradiction that


Then there exists   such that for all  


since   is continuous. (At some point,   will either monotonically increase or decrease to  .) This implies


which contradicts the hypothesis that   is integrable i.e.  . Hence,


Using the same reasoning as above,




Alternate Solution edit

Suppose   (without loss of generality,  ). Then for small positive  , there exists some real   such that for all   we have  . By the fundamental theorem of calculus, this gives

  for all  .

Since   is integrable, this means that for any small positive  , there exists an   such that for all  , we have  . But by the above estimate,


This contradicts the integrability of  . Therefore, we must have  .

Problem 3 edit

Suppose that   is a sequence of real valued measurable functions defined on the interval   and suppose that   for almost every  . Let   and   and suppose that   for all  

(a) Prove that  .

(b)Prove that   as  

Solution 3a edit

By definition of norm,


Since  ,


By Fatou's Lemma,


which implies, by taking the  th root,


Solution 3b edit

By Holder's Inequality, for all   that are measurable,




The Vitali Convergence Theorem then implies


Problem 5 edit

Suppose  . Prove that   and that


Solution 5 edit