2009 Spring Real Analysis

Problem 1.

Let be a measure space. Let and be real-valued -measurable functions on a set . Suppose on , that is, converges to in measure on . Let be a real-valued uniformly continuous function on . Show that on .

Proof.

Since is uniformly continuous, for any , there is a such that whenever , we have . Thus we have

Thus if is convergent to in measure, then is convergent to in measure.

Problem 2.

(a) Let , . Prove that .

(b) Suppose . Prove that .

Proof.

This is very similar to Problem 2 of 2017 Fall Real Analysis Exam.

Problem 3.

Let . Prove that for ,

Proof.

We have

For any , let such that

Let . We have

On the other hand, since is smooth with compact support, it must be uniformly continuous. Thus for large enough, we must have

where . Then

Using triangle inequality, we have

for .

Problem 4.

Suppose is a bounded nonnegative function on with . Show that is integrable if and only if

Proof.

This is very similar to Problem 1 of 2006 Fall Real Analysis Exam.

Problem 5.

Let be an element and be a sequence in where such that . Show that for every there exists such that for all we have

Proof.

By triangle inequality, we have

By Dominated convergence theorem, for any there is a such that if , we must have

This completes the proof.

Problem 6.

Let be a measure space with , and let be a sub--algebra of . Given an integrable function on , show that there is a -measurable function , such that

for every -measurable function such that is integrable. (Hint: Use the Radon-Nikodym Theorem.)

Proof.

This is the same as Problem 1 of 2016 Fall Real Analysis Exam.