2021 Spring Real Analysis

Problem 1.

Let be the Lebesgue measure on and be the counting measure. Show that does not have a Lebesgue decomposition with respect to .

Proof.

Assume such that

Then by the Lebesgue decomposition theorem must be the counting measure plus a singular measure. Since is the counting measure it follows that has non-zero component which is a singular measure, which cannot be absolutely continuous with respect to the Lebesgue measure.

Problem 2.

Let be a (not necessarily measurable) set with , where is the Lebesgue outer measure given by

and denotes the length of an interval . Show that

Proof.

Assume there exists and such that for all intervals . Let be small enough such that

Then

On the other hand, we have

Then and so . Letting we get , a contradiction.

Problem 3.

Let be a measure space and prove that there exists a function satisfying if and only if is -finite.

Proof.

Let

Then .

Thus if , then is -finite.

On the other hand, if is -finite, we write

where . Define

for . Then with .

Problem 4.

Let be a probability space, that is, a measure space with . Let be a strictly positive -measurable function on . Prove that

Proof.

For any positive function we have

that is

Let . We have

By choosing

we get

completing the proof.

Remark.

See the remark of Problem 5 of 2023 Winter Real Analysis Exam for the similar trick.

Problem 5.

Suppose that and prove that

Proof.

This problem is very similar to Problem 5 of 2018 Fall Real Analysis Exam.

we have

Observe that

for some constant . Thus

In particular, for almost all ,

Since are arbitrary, the theorem is proved.

Problem 6.

Let and be two measurable subsets of with finite measures and define

for the Lebesgue measure . Prove that

justifying all the steps in your calculation.

Proof.

Let be the characteristic function of a set . Then the function .

Then

by Fubini's theorem, and hence

by the change of variable formula.