2016 Spring Real Analysis

Problem 1.

Assume that . Compute

Justify your answer.

Proof.

Let

We claim that

It is obvious that, point-wisely, that

for all . Moreover, we have

Since is integrable, by using the Dominated convergence theorem, we conclude the claim.

Problem 2.

Let be a sequence of measurable functions on and a.e. Assume that

for some and any . Prove that a.e..

Proof.

For each , define

For any , let such that

Thus we have

Letting , we get

Since is arbitrary, this implies =0$. As a result, we have

and implies that a.e.. Similarly, we can prove that a.e..

Problem 3.

Let . Show that is a continuous function on vanishing at infinity, that is and .

Proof.

See Problem 4 of 2014 Spring Real Analysis Exam.

Problem 4.

Let be a finite measure space, and let . Let be a uniformly bounded sequence in , that is, and . Suppose exists -a.e. Prove that for all and in for all .

Proof.

Since is a finite measure space, is bounded in for all by Hölder's inequality. By Fatou's lemma, we have

Finally, by Egorov's theorem, for any , there is a set with and on , the convergence is uniform. For , by triangle inequality, we have

By Hölder's inequality, we have

By the uniform convergence, there is an such that for , we have

Thus we have

and this completes the proof.

Problem 5.

Let be a measure space, and let be -measurable. Consider the measure space , where is the Borel -algebra on and is Lebesgue measure, and form the product measure space . Define by . Prove that and .

Proof.

We let be a sequence of increasing simply functions such that . Then

where

Thus is measurable with respect to the product measure. Moreover we have

Problem 6.

Let , and let and . Assume that the vectors are all distinct. Determine

Proof.

For any , let be a smooth function such that

Assume the support of is within for some , then the supports of must be within

Since are distinct, for , these intervals are disjoint. Therefore we have

Finally, by (1), we have

Thus