2014 Fall Real Analysis

Let be the collection of all subsets of that consist of exactly 5 points. Find the -algebra of sets generated by .

It contain the empty set, , any sequence ,and the complement of any sequence.

Assume that is a non-negative real valued function satisfying . Show that

This follows from the obvious inequality

Denote

Show that .

Define

for pairs of integers and let be the lexicographic order of . Then we can write

Obviously, . Thus

Note that for any , there is at most one satisfies the inequality , we have

As a result, we conclude that .

Assume that is a non-negative function satisfying . Show that for any ,

Here

This is a simple application of the Tonelli's theorem.

Let be continuous and periodic with period one. Prove that

This is the Riemann–Lebesgue lemma. A simple way to prove this problem in the context is as follows. Let

Then is a periodic function with period and . As a result, there is a periodic function of period such that . Note that in this case,

as . Thus

Let be the decimal representation of . Compute with justification , where

(i) ;

(ii) .

(i). I claim that a.e.. To see this, let . If , then

for any . Thus

Since

we have

As a result, we have

This proves that is almost everywhere equal to , and hence $\int_0^1 f(x) dx=9.

(ii). We use a similar method here. We claim that almost everywhere. To see it, let be a large number. Define

Then

As a result,

This means, for almost all , there is a sequence of decimal of length . Thus

Since is arbitrary, we get a.e. On the other hand, by definition. Thus a.e., and hence .