2011 Spring Real Analysis

Problem 1.

Suppose and are real-valued -measurable functions on , such that (1) is -integrable, and (2) belongs to . For define . Prove that

(a) , and

(b) .

Proof.

Assume that for some number . Then

Thus we have

where . The result follows from the Dominated convergence theorem.

On the other hand, for any , there is an such that

For such an , if is small enough, we have for all . Thus

completing the proof.

Problem 2.

Suppose and are -finite measures on a measurable space , such that and . Prove that

Proof.

Let

Then for any measurable set , we have

Let be the set over which . Then we have

Problem 3.

Prove that the Gamma function

is well defined and continuous for .

Proof.

This improper integral has two singularities. If , then is integrable at . On the other hand, if , . So the function is also integrable at .

Problem 4.

Let and be the measure spaces given by

.
the Borel -algebra of .
, and is the counting measure.

Consider the product measurable space , and its subset .

(a) Show that .

(b) Show that .

Proof.

This is the same as Problem 3 of 2021 Winter Real Analysis Exam.

Problem 5.

Suppose (that is, is coninuous and continuously differentiable on ), , and everywhere.

(a) Prove that everywhere.

(b) Prove that

Hint: Apply Cauchy-Schwarz (or Hölder) Inequality to the product of and , where , and .

Proof.

By assumption, we know that is monotonically decreasing. Thus . As a result, we must have . Hence . This proves the positivity of .

Using the Cauchy inequality, we have

The theorem is proved by observing that

Problem 6.

Suppose is a sequence of measurable functions on . For define (the number of indices for which ). Assuming that everywhere, prove that the function is measurable.

Proof.

Let for all . Since is measurable, so is . Note that

so it has to be measurable.