2010 Spring Real Analysis

Problem 1.

Suppose is Lipschitz continuous in , that is, for some constant . Show that

(a) if .

(b) If is measurable, then is also measurable.

Proof.

(a). For any , there exists an open set

such that

It is not hard to see that

Thus by taking limit.

(b). Every measurable set is the union of an set and a Lebesgue set. An set is a countable union of bounded closed set (=compact set). Since is continuous, it maps a compact set to a compact set, and this it maps an set to an set. Combining this with (a), can be expressed as a union of an set to a Lebesgue measure zero set, hence measurable.

Problem 2.

Let be all the rational numbers in . Show that

Proof.

This is very similar to Probloem 4 of 2017 Fall Real Analysis Exam.

Problem 3.

Suppose that where is integrable over . Show that

Proof.

We have

We claim that

as . If not, then there an such that we can find sequences such that

where and . By the construction of the closed intervals , we know that either one contains the other, or two of them intersect at most at one point. Since is integrable, we can find an interval such that it contains infinitely many intervals of the form . Inductively, we are able to find a subsequence such that

By the Dominated convergence theorem, we have

a contradiction.

Problem 4.

Let be a real-valued uniformly continuous function on . Show that if is Lebesgue integrable on , then .

Proof.

If not, then there is an and sequence such that

By choosing a subsequence if necessary, we may assume that . By uniform continuity, there is a such that whenever . Thus . As a result, we have

Assuming that , we have

which is a contradiction.

Problem 5.

Let be a measure space and let be an extended real-valued -measurable function on such that for some . Show that

Proof.

We have

By the above first inequality, we have

Applying the Dominated convergence theorem](https://en.wikipedia.org/wiki/Dominated_convergence_theorem), we know that the above right side tends to zero as .

Problem 6.

Consider the Lebesgue measurable space on . Let be a -integrable extended real-valued -measurable function on . Show that

Proof.

This is very similar to Problem 5 of 2022 Fall Real Analysis Exam.