2022 Fall Real Analysis

Problem 1.

Let be a msasure space and a sequence in that converges a.e. to . Prove that in as if and only

as .

Proof.

The only if part follows from the triangle inequality

To prove the if part, we first do some preparation. Let be a subset. Then we have

We also have

Combining the above two inequalities, we have

Now we prove the if part of this problem. For any , we can find a subset such that

and . By Egorov's Theorem, there exists a set such that

and on , uniformly. Now let , then using (1), we have

The theorem follows from the assumption and the uniform convergence of over .

Problem 2.

Suppose that is a probability space and is a -subalgebra of . Further suppose that .

(a). Prove that there is a unique such that for all .

(b). Suppose that for some and that for some with . Write an explicit formula for the function from part (a) in this case. Make sure to justify your answer.

Proof.

For part (a), we define

for . It is not hard to see that , that is, is absolutely continuous with respect to . Then by Radon–Nikodym theorem, there is a measureable function such that

For part (b), we define

for and

for . Then is the Radon-Nikodym derivative of to .

Problem 3.

Let , , and assume that

for any . Conclude that a.e..

(Hint: Show that the inequality extends to characteristic functions of balls.)

Proof.

Let be a ball of radius centered at . Let be the characteristic function of . Then for any , there is a smooth function with compact support such that on , and otherwise. Thus we have

We also have

Thus we have

Letting , we obtain

This means that the inequality extends to characteristic function of balls. Rewrite the above inequality as

By the Lebesgue Differentiation Theorem, we conclude that a.e..

Problem 4.

Let be finite signed measure on a measurable space . Show that

for . Also prove the inequality can be strict.

Proof.

The inequality is obvious. To prove that strict inequality can occur, take .

Problem 5.

Let and show that

Proof.

This is very similar to Problem 4 of 2023 Winter Real Analysis Exam. The proof essentially goes as follows: first prove the theorem for smooth functions with compact support. Then use smooth functions to approximate functions.

Problem 6.

Let be the Lebesgue measurable sets. Suppose that each belongs to at least 5 of these sets (which 5 can vary with ). Prove that for some , where is the Lebesgue measure on the real line.

Proof.

This is very similar to Problem 1 of 2023 Winter Real Analysis. Let be the characteristic function of . By assumption, we have

Integrating on both sides, we obtain

Thus there is an such that .