2023 Spring Real Analysis

Problem 1.

Let be a measure space and let be a measurable function.

(a). Prove that, for any , one has

(b). Suppose that and there is such that

for every . Prove that .

Proof.

It is well-known that

Thus (a) follows from change of variable .

Using (a), we have

Since , we have

Problem 2.

Recall that the Cantor-Lebesgue function is the function first defined on the Cantor set by setting , where and , and then continuously extended to all of by setting it to be constant on the intervals deleted in the formation of the Cantor set. Let be the Borel measure on defined by for all . Prove that is not absolutely continuous with respect to Lebesgue measure.

Proof.

An absolutely continuous function does not fluctuate on a set of measure zero. But a Cantor set is of measure zero. To see that the Cantor-Lebesgue function is not absolutely continuous, we take any . Let the set of intervals covering the Cantor set but

Then we get a contradiction since

is the total variation of the function.

Problem 3.

Suppose that is integrable and is bounded and measurable. Prove that

Proof.

We can prove an even stronger result

The problem is similar to Problem 5 of 2022 Fall Real Analysis Exam.

Problem 4.

Suppose that is nonnegative. Prove that

Proof.

We observe that for any ,

which is integrable. On the other hand,

if and is equal to if . Thus the conclusion follows from the Lebesgue dominant convergence theorem.

Problem 5.

Let be a Lebesgue measurable set such that for each . Prove that or , where denotes Lebesgue measure.

Proof.

If the measure of is positive, then by Lebesgue differentiation the density of in some interval is at least very close to . Using that translations by a dense set of this interval are contained in , we see that this density lower bound holds in arbitrary sets. Then take to .

Problem 6.

Fix a measure space , , and functions such that almost everywhere. Prove that the following two conditions are equivalent:

(a). in .

(b). .

Proof.

This problem is very similar to Problem 1 of 2022 Fall Real Analysis Exam.