2010 Fall Real Analysis

Problem 1.

Consider a measure space and a sequence of measurable sets , , such that . Show that almost every is an element of at most finitely many .

Proof.

This is the same as Problem 1 of 2023 Winter Real Analysis Exam.

Problem 2.

Consider a measure space with , and a sequence of measurable functions such that for all . Show that for every there exists a set of measure such that converges uniformly to outside the set .

Proof.

This is the Egorov's theorem. The key is to write

Thus we have

In other words, for any , we have

Let b be any small positive number. Since , then for each , there is an such that

It follows that

Since

is convergent to uniformly on .

Problem 3.

Suppose that for some . Prove that .

Proof.

We have

By Hölder's inequality, we have

where . Therefore is integrable and this completes the proof.

Problem 4.

Assume is measurable and for any ,

Show that .

Proof.

This is very similar to Problem 4 of 2022 Spring Real Analysis Exam. Let . Then we have

Thus on any measurable set ,

As a result, we know that a.e.. Thus for a.e. , . So .

Problem 5.

Let be a real-valued uniformly continuous function on . Show that if is Lebesgue integrable on , then .

Proof.

This is the same as Problem 4 of 2010 Spring Real Analysis Exam.

Problem 6.

Consider the Lebesgue measure space on . Let be an extended real-valued -measurable function on . For and let .

With fixed, define a function on by setting

(a) Suppose is locally -integrable on , that is, is -integrable on every bounded -measurable set in . Show that is a real-valued continuous function on .

(b) Show that if is -integrable on then is uniformly continuous on .

Proof.

(a). Let , we can write

Continuity follows from the local integrability and the Dominated convergence theorem.

(b). To prove the uniform continuity, we just need to note that since is in , then as . Any continuous function on with limit exists at infinity must be uniformly continuous. One of the proof is as follows: compactify by adding the infinity point to get a compact space . The limit at infinity exists means that the function can be extended as a continuous function on . Since is a compact space, any continuous function on it must be uniformly continuous.