2006 Spring Real Analysis

Problem 1.

Let with . Prove that for all .

Proof.

See Problem 5 of 2023 Winter Real Analysis Exam.

Problem 2.

Let be a Lebesgue measurable sets on the real line. Consider two measures on : Lebesgue measure and the counting measure , where for we set to be the number of points in . Show that is absolutely continuous with respect to , but that doesn't exist, i.e., there is no measureable function such that for all . Does this contradict to the Radon-Nikodym Theorem?

Proof.

See Radon-Nikodym theorem. The Radon-Nikodym derivative doesn't exist because:

, live in two different -algebra;
is not -finite.

Problem 3.

Given a measure space , let and be extended real-valued -measurable functions on a set and assume that is real-valued a.e. on . Suppose there exists a sequence of positive numbers such that:

for every for some fixed

Show that the sequence convergens to a.e. on .

Proof.

This problem is similar to Problem 4 of 2023 Winter Real Analysis Exam and Problem 2 of 2014 Spring Real Analysis Exam. Here we provide a proof for the sake of completeness.

Define

Then from,

we conclude that . It follows that

and this completes the proof.

Problem 4.

Let be a real-valued function of bounded variation on . Suppose is absolutely continuous on for every . Show that is absolutely continuous on .

Proof.

（This problem is wrong.)

Problem 5.

Let be a collection of pairwise disjoint subsets of a -finite measure space, each of positive measure. Show that is at most countable.

Proof.

We just need to prove that the number of subsets of measure greater than is coutable. If the total measure of the space is finite, then this number is finite. If the space is -finite, then this number is at most countable.

Problem 6.

Let be a complete measure space and let be a nonnegative integrable function on . Let . Show that:

Proof.

Let

Then

Using Fubini's Theorem, we have: