2017 Fall Real Analysis

Problem 1.

Assume where is the Lebesgue measure. Let . Prove or disprove:

(a) as ;

(b) as ;

Proof.

(a) follows from

(b) Since as , we must have

as . On the other hand, we have

Thus

as . Thus (b) is also true.

Then , but there is no such zero measure set .

Problem 2.

Let be a Lebesgue measurable extended real valued function on . Let and , and suppose we have . Prove that

Proof.

We use Hölder's inequality to get

The theorem follows.

Problem 3.

Consider the Lebesgue measure in . Suppose is measurable. Let

Show that if for -a.e. , then

Proof.

See Problem 4 of 2015 Spring Real Analysis Exam.

Problem 4.

Show that for a.e. and for every

Proof.

We observe that

Thus

The theorem follows.

Problem 5.

Let be a measure on that is absolutely continuous with respect to the Lebesgue measure and with Radon-Nikodym derivative equal to . Assume that are -measurable functions with for all . Prove or disprove that for Lebesgue a.e. ,

(a) ;

(b) .

Proof.

That we use a measure and being non-compcact is irrelevant. This problem is similar to Problem 4 of 2023 Winter Real Analysis Exam, Problem 2 of 2014 Spring Real Analysis Exam.

Problem 6.

Let be a measure space and . If -a.e., show that there exist sets such that , and uniformly on each .

Proof.

This is essentially the Egorov's theorem. For each , we let be a set such that uniformly but . Let

Then , and we have