2015 Spring Real Analysis

Problem 1.

Show that if then

Proof.

The is the same Problem 5 of 2022 Fall Real Analysis Exam.

Problem 2.

Let , be a sequence of Lebesgue measurable functions such that a.e. as . Assume that there exists such that , and for all . Assume furthermore that

for some constant . Show that , and in as .

Proof.

Equation (1) is equivalent to, after change of variable,

Thus for any , we have

Let , we choose such that

Since on , is integrable, by the Dominated convergence theorem, there is an such that for any , we have

On the other hand, by the Fatou's lemma,

It follows that

and the theorem is proved.

Problem 3.

Assume that and

Show that exists.

Proof.

Let . Then by Hölder's inequality, we have

where . Since

as , exists by the Cahchy criterion.

Problem 4.

Suppose that is measurable. Denote

Show that if for a.e. , then

Proof.

By Fubini's theorem, we have

If for a.e. , then

which implies the conclusion.

Problem 5.

Let , , and let . Show that the series

converges for a.e. .

Proof.

We assume that is nonnegative without loss of generality. Let . We consider

Since , we have

By Young's inequality, we have

By assumption on , we know that . Thus the second term in the above right is convergent. For the first term in the above right, we are in the same situation as in Problem 5 of 2021 Spring Real Analysis Exam and Problem 5 of 2018 Fall Real Analysis Exam, and we can conclude that it is also bounded:

We thus conclude that

is integrable. The theorem thus follows.

Problem 6.

Suppose is measurable and for every natural number . Show that either or .

Proof.

Let be any real number. Let be the smallest positive integer such that

Inductively we define such that it is the smallest positive integer such that

for any . We claim that

If not, then

In order to prove the claim, we notice that by construction of , for each , we either have , or

Since the harmonic series is divergent, is not true for all sufficient large . As a result, we shall find a subsequence such that

and

Thus and this is a contradiction. The claim is proved.

By the claim, we have for all . It is not hard to extend the same translation invariance to all real numbers . We are then in the same situation as in the Problem 5 of 2022 Winter Real Analysis Exam and we can use the method there to complete the proof.