2008 Fall Real Analysis

Problem 1.

Let be a Lebesgue integrable function of the real line. Prove that

Proof.

Let such that

We compute that

Thus

The theorem follows.

Problem 2.

Let be an absolutely continuous monotone function on . Prove that, if is a set of Lebesgue measure zero, then the set is also a set of Lebegue measure zero.

Proof.

This essentially follows from the absolute continuity. Assume that is monotonically increasing. Let , then there is a such that if is an open set with , and we write

Then

However, since

we know that for any , concluding the theorem.

Problem 3.

Let be a finite Borel measure on the real line, and set . Prove that is absolutely continuous with respect to the Lebesgue measure if and only if is an absolutely continuous function. In this case show that its Radon-Nikodym derivative is the derivative of , that is, almost everywhere.

Proof.

This problem is almost the same as the above problem.

Problem 4.

Let be a measure and let , , be signed measures on the measurable space . Prove:

(a) If and then .

(b) If and , then, if we set with , real numbers such that is a signed measure, we have .

Proof.

(a). Since , there is a set such that and . However, since , for any subset , we have . Thus .

(b), (c) are obvious.

Problem 5.

Let be a measure space. Let and be extended real-valued -measurable functions on a set such that on . Then for every we have

(a) .

(b) .

Proof.

We first note that (b) follows from (a) by replacing by ; by , and by . So we just need to prove (a).

For (a), without loss of generality, we assume that is of finite measure. Let . by Egorov's theorem, for any , there is a set such that and uniformly on . As a result, there exists an such that for

for all . As a result,

Since both can be made arbitarily small, (a) follows.

Problem 6.

Let be a measure space. Let and be a sequence of extended real-valued -measurable functions on a set with . Show that converges to 0 in measure on if and only if .

Proof.

This is the same as Problem 2 of 2020 Fall Real Analysis Exam.