UMD Analysis Qualifying Exam/Aug06 Real
Contents
Problem 1a[edit]
Prove the following version of the RiemannLebesque Lemma: Let . Prove in detail that

Solution 1a[edit]
Note that .
Hence we can equivalently show
as
Claim[edit]
Let be a step function.
as
Proof[edit]
Step functions approximate L^1 functions well[edit]
Since , then
Hence, given , there exists such that
Problem 1b[edit]
Let be an increasing sequence of positive integers. Show that has measure 0.

Solution 1b[edit]
Problem 3[edit]
Suppose , where . Show that . 
Solution 3[edit]
Let then we can write
Hence .
Problem 5[edit]
Let , (a) Show that is differentiable a.e. and find . (b) Is absolutely continuous on closed bounded intervals ? 
Solution 5[edit]
Look at the difference quotient:
We can justify bringing the limit inside the integral. This is because for every , . Hence, our integrand is bounded by and hence is for all . Then by Lebesgue Dominated Convergence, we can take the pointwise limit of the integrand. to get
It is easy to show that is bounded (specifically by ) which implies that is Lipschitz continuous which implies that it is absolutely continuous.