2023 Fall Real Analysis

Problem 1.

Let be a positive continuous function on . Prove that

Proof.

First note that

Consequently, .

On the other hand, suppose that . (If , then is constantly and the result is trivial.) Set ; since , we have that . Now note that

It follows that Since was arbitrary, we get .

The two inequalities together yield the desired conclusion.

Problem 2.

Suppose that is absolutely continuous and is absolutely continuous and strictly monotonic. Prove that is absolutely continuous.

Proof.

Fix and take as in the definition of absolute continuity for for this given . Now take as in the definition of absolute continuity for for this given . We claim that this witnesses the absolute continuity for for the given . To see this, suppose that , , is a collection of pairwise disjoint open intervals such that Since is strictly monotonic and continuous, we see that is a collection of pairwise disjoint open intervals; moreover, by the choice of , we have that . By the choice of , we have that , as desired.

Problem 3.

Suppose that is such that

for every absolutely continuous function on for which . Prove that equals to a constant almost everywhere.

Proof.

By the Lebesgue differentiation theorem, it suffices to show that takes the same value at all its Lebesgue points in . Let and be any two Lebesgue points of in with . For small, let be the Lipschitz (hence absolutely continuous) function that is zero outside of , one in , and linear on the remaining intervals of length centered at and . Then

Letting go to zero (and recalling that and are Lebesgue points of ), we conclude that , as desired.

Problem 4.

Fix and . Let be the function on defined by

Prove that is continuous.

Proof.

Let . Applying Hölder's inequality to the function , we reduce the problem to showing that as . For , let be a continuous, compactly supported function for which . Let . Then

and the right hand side is smaller than for sufficiently small since is continuous.

Problem 5.

Fix and . Show that

Proof.

If , then

If , we use Young's inequality

where . Then

and we conclude by noting that the right hand side of this inequality is integrable.

Problem 6.

Fix . Show that the series

converges absolutely for almost every .

Proof.

For any set , let denote its characteristic function. Let

For any , we have

If , then

Thus is bounded for a constant . Therefore,

As a result, for almost all , is absolutely convergent.