2014 Spring Real Analysis

Problem 1.

Let be a subset of of positive Lebesgue measure. Prove that there exists and with .

Proof.

For any , define

If the statement is not true, then for any , we have . Moreover, if , then . Let be an interval such that . Then for any , we must have

Since

we get a contradiction, because the left side of the above is no less than infinitely many sum of the same positive number .

Problem 2.

Either prove or give a counterexample: if a sequence of functions on a measure space satisfies

then almost everywhere.

Proof.

For any define

Then if , for any , we must have

As a result, we have

In particular, this implies that for any . Thus we have

Hence for almost all , .

Problem 3.

Let , and let

Show that

for all .

Proof.

Using Hölder's inequality, we have

Thus we have

where is the characteristic function of the interval which is on the interval and outside. Since for fixed , we have

almost everywhere, by the Lebesgue's dominated convergence theorem, we get

Problem 4.

Assume that . define

Show that and

Proof.

Let be a small number, we then have

by changing of variable. Thus we have

Using Cauchy inequality we obtain

Since , we have

for a constant . Thus we have

It remains to prove that

For any , there exists an smooth function such that

For any , we shall also have

by the change of variable formula. Since is a smooth function with compact support, its derivative is bounded by a constant . Assume that the support of is contained in . If we choose small enough, we have By triangle inequality, we have and this proves the continuity of .

To prove that tends to at infinity, we assume that be a small number. By the integrability of , there exists an such that

Then for any , we have

For the second term in the above right, we have

Since

we have

Similarly, we get the estimate of the first term by the following conversion:

Problem 5.

Is it possible for a continuous function to have

(a). Infinitely many strict local minima?

(b). uncountably many strict local minima?

Proof.

For question (a), the function is an example of infinitely many strictly local minima at

It is not possible to have uncountably many strictly local minimal for a function. Let be a strictly local minimal point. Then there exists a number such that for any , we must have . We assume is the maximal number such a property holds.

Let be the set of strictly local minimal points. Let

Then

If is uncountable, then at lease one of must be uncountable, and hence infinite on some closed interval . However, since for any , we must have , the number must be finite. This is a contradiction.

Problem 6.

Let be the collection of functions such that and . Prove that for every ,

If you haven't seen , it is defined in a natural way, or else as

Proof.

We writre as the positive and negative part of . Then we have

Thus we have

As a result, we have

and hence

To prove that the above inequality is actually an equality, we just need to let be (half) Dirac delta functions on the maximal and minimal points of .