2021 Winter Real Analysis

Problem 1.

(a) Give an example of a sequence of Lebesgue measurable functions on that does not converge to zero almost everywhere, but nonetheless

(b) Is there such an example if one assumes that

Give an example or prove there isn't one.

Proof.

Problem 2.

(a) Give an example of a measure on the unit interval that differs from Lebesgue measure such that:

(i) ;

(ii) Every Borel set is -measurable;

(iii) If , then ;

(iv) If and then .

(b) Show that for all measures on the unit interval satisfying properties (i-iv) there is a measure on the unit square such that

(i) ;

(ii) Every Borel set is -measurable;

(iii) If then ;

(iv) For

Define the measure and prove it has properties (i-iv).

Proof.

For (a), we take any positive continuous function such that

Then the measure satisfies all the requirements.

For (b), we take

Problem 3.

Let and be the measure spaces given by given by:

;
, the Borel -algebra of ;
(Lebesgue measure) and is the counting measure.

Consider the product measure space and a subset of it defined by . Then show that

(a) ;

(b) ;

Proof.

(a). Since

is the countable intersection of open sets, it is measurable.

(b) We have

but

So they are not equal.

(c).This is because with respect to the counting measure, is not -finite.

Problem 4.

Let and be -finite measure spaces, and consider the product measure space . Let be a real-valued, -measurable and -integrable function on and a real-valued, -measurable and -integrable function on . Consider the real-valued function on given by . Show that is a -measurable and -integrable function on , and

Proof.

This follows from approximating both and by simple functions.

Problem 5.

is a measurable function and is non-empty open subset of . Let . Define

Show that is a real-valued measurable function.

Proof.

We write . Then on , if and if . In either case, is measurable on .

We then only need to prove that is measurable on . If , then and is measurable. So for the rest of the proof, we assume that . Write

as disjoint union of open intervals. Then

So it suffices to prove that

is a measurable function.

Let and be a sequence of real numbers such that on is increasing and on is decreasing. Let for all . Note that is never zero. Define if has no solution and if . Then are measurable functions. Since , it must be measurable. This completes the proof.

Problem 6.

is a sequence of real valued function in . Suppose that for any ,

Prove that

(You may NOT simply quote the Banach-Steinhaus Theorem or the Uniform Boundedness Principle.)

Proof.

For any , let

By assumption, we have

Since is a complete metric space, it is of second category. By the Baire category theorem, at lease one of must be of second category. Thus for such an , is dense on a ball for some and some . By continuity. we know that for any , we have

Now let . Write for some with . Then

where is defined to be

Replacing by for , we get

Letting , we shall get

which implies the theorem.