7.2.E: Problems on -Sets, -Additivity, and Permutable Series

Exercise

Fill in the missing details in the proofs of this section.

Exercise

Prove Note 3.

Exercise

Show that every open set in is a countable union of disjoint half-open cubes.
[Outline: For each natural show that is split into such cubes of edge length by the hyperplanes

and that the family of such cubes is countable.
For let be the sequence of those cubes from (if any) that lie in but not in any cube with

Exercise

Prove that any open set is a countable union of disjoint (possibly infinite) open intervals.
[Hint: By Lemma If, say, overlaps with some , replace both by their union. Continue inductively.

Exercise

Prove that is closed under finite intersections and countable unions.

Exercise

(i) Find such that
(ii) Show that is not a semiring.
[Hint: Try (the rationals).]
Note. In the following problems, is countably infinite,

Exercise

Prove that

iff for every there is a finite set

such that

for every finite .
[Outline: By Theorem 2, fix with

By Cauchy's criterion,

iff

Let If is as above,

so

Exercise

Prove that if

then for every there is a finite such that

for each finite .
[Hint: Proceed as in Problem with and so large that

Exercise

Show that if

then

(Use Problem 8' below.)

Exercise

Show that

over all finite sets .

Exercise

Show that if then

Exercise

Continuing Problem prove that if

then

[Outline: By Problem 9,

so

and

converge absolutely.
Fix and as in Problem Choose the largest with

(why does it exist?), and fix any By Problem

(Explain!) Let

so

and By Problem 7,

Deduce

Exercise

(Double series.) Prove that if one of the expressions

is finite, so are the other two, and

with all series involved absolutely convergent.
[Hint: Use Problems 8 and with

so

Thus obtain

Similarly,