2006 Fall Real Analysis

Problem 1.

Given a measure space , let be a nonnegative extended real-valued -measurable function on a set with . Let for . Show that if and only if .

Proof.

We have the following inequalities

By Abel's summation formula, we have

Thus if , then . On the other hand, if , then

We thus have

Problem 2.

Let be a -finite measure space and let and be extended real-valued -measurable functions on . Show that if for every then a.e. on .

Proof.

Since for all measurable set , we get the conclusion by approximating by simple functions.

Problem 3.

Let be an integrable function on a measure space . Show that:

(a) The set is of -finite measure.

(b) If , then pointwise for some increasing sequence of simple functions each of which vanishes outside a set of finite measure.

Proof.

(a). For any , we have

Thus the set is of finite measure, and hence

is -finite.

(b), (c) I don't know what to prove?

Problem 4.

Let and be sequences of measurable functions on the measurable set , and let and be measurable functions on set . Suppose and in measure. Is it true that in measure if

(a) .

(b) .

Proof.

(a). It is true. Let . Then as . For any , we can find an such that . For this , if , we must have

Thus we have

Thus if , then the above measure goes to zero.

(b). It is not true. Let . Let . . Apparently in measure. But on the other hand, if ,

Thus doesn't convergent to in measure.

Problem 5.

Suppose . Let . Let be a Lipshitz function (i.e. for some ). Show that there exists such that .

Proof.

We need to prove that is absolutely continuous with respect to the Lebesgue measure. For any , there exists a such that for any sequence of real numbers

with , we have

where . We thus have

This is absolutely continuous, and hence in question exists.

Problem 6.

(a) Suppose , . Show that .

(b) Suppose . Show that .

Proof.

We use Hölder's inequality to prove both problems.

(a). We have

Thus

By Dominated convergence theorem, we have

(b). Using Hölder's inequality, we have

By Dominated convergence theorem, we prove the result.