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2012 Fall Real Analysis

Problem 1.

Find

(a) ;

(b) .

Justify your answers.

Proof.

BY the Dominated convergence theorem, we have

(a).

(b).

Problem 2.

Find a sequence of pointwise convergent measurable functions such that doesn't converge uniformly on any set with .

Proof.

I don't quite understand this problem. Apparently is convergent to pointwise but not uniform on .

Problem 3.

Suppose is Borel measurable and

Show that .

Proof.

This is very similar to Problem 6 of 2006 Fall Real Analysis Exam.

Problem 4.

If is a finite measure space, , , and

prove that .

Proof.

Let . Then the condition is equivalent to

By Problem 6 of 2006 Spring Real Analysis Exam, we have

Problem 5.

Let be a decimal representation of . Let . Prove that is measurable and a.e. constant.

Proof.

Define

Then . Thus

if , then . Thus is a.e. .

Problem 6.

Let be a sequence of measurable functions on with and . Show that

Proof.

Let . Define

Since , we must have

As a result, we have

and hence .

For any , we shall have

Since is arbitrary, the result is proved.