2007 Fall Real Analysis

Problem 1.

Let . Which of the following statements are true and which are false? Justify.

(a) .

(b) .

(d) .

Proof.

This is very similar to Problem 1 of 2019 Fall Real Analysis Exam.

Problem 2.

Does there exist a Lebesgue measurable subset of such that for every interval we have ? Either construct such a set or prove it does not exist.

Proof.

This is the same as Problem 4 of 2022 Spring Real Analysis Exam.

Problem 3.

Let be a real valued measurable function on the finite measure space . Prove that the function is measurable in the product measure space , and that is integrable if and only if is integrable.

Proof.

This is the same as Problem 6 of 2011 Fall Real Analysis Exam.

Problem 4.

Let . Prove that

Proof.

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

Problem 5.

Let be a sequence of real-valued functions and be a real-valued function on such that for . Let be the total variation of on . Show that

Proof.

This is the same as Problem 1 of 2022 Winter Real Analysis Exam.

Problem 6.

Let . With fixed, define a function on by setting

(a) Show that is -measurable on .

(b) Show that and .

Proof.

(a). is obvious. In fact, is absolutely continuous.

(b). is very similar to Problem 3 of 2009 Spring Real Analysis Exam.