2022 Winter Real Analysis

Problem 1.

Let converge pointwise to . Show that

Before embarking on the proof, state the definition of TV (total variation).

Proof.

Let be a sequence of numbers in . Define the total variation of to be

Now we prove the theorem. By assumption, we have, for any and , that

Taking supremium of the left side above, we get the conclusion. ghg

Problem 2.

Let and be two positive measures on a measurable space . Suppose that, for every , there exists such that and . Show that .

Proof.

For any , we can find such that and . Let

By definition, we have

for any , and thus

As a result, we have

On the the hand, we consider

We have

Thus . This completes the proof.

Problem 3.

Let be a probability measure, i.e. a measure on with . Show that, given measuable , it holds that

Proof.

This problem is very similar to Problem 1 of 2023 Winter Real Analysis Exam and Problem 6 of 2022 Fall Real Analysis Exam. Let

where is the characteristic function of . Then is a positive integer valued function. We have

Thus the measure must be positive. Then obviously

Problem 4.

Let be a measure space, and let . Let be a sequence of -measureable sets such that

Prove that

Proof.

We can write so it suffices to prove the theorem assuming that is nonnegative. We observe that

The sequence of functions almost everywhere by the assumption of the measures of . Then by Fatou's lemma, we have

This completes the proof.

Problem 5.

Suppose that is a measurable set of positive measure and be a dense set. Prove that

is a set of measure zero.

Proof.

We shall use the Lebesgue's Density Theorem. Assume that the masure of is positive. Then there is a point such that for any , there exists an such that . Since is of positive measure, there exists a such that . Choose some such that . This is not possible since .

Problem 6.

Suppose that and prove that

Proof.

For any , there exists an such that

for . For such a fixed , we let large enough so that

Thus we have

This completes the proof.