UMD Analysis Qualifying Exam/Aug06 Real
Prove the following version of the Riemann-Lebesque Lemma: Let . Prove in detail that
Note that .
Hence we can equivalently show
Let be a step function.
Step functions approximate L^1 functions well
Since , then
Hence, given , there exists such that
Let be an increasing sequence of positive integers. Show that has measure 0.
Suppose , where . Show that .
Let then we can write
(a) Show that is differentiable a.e. and find .
(b) Is absolutely continuous on closed bounded intervals ?
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.