
# 7.2.E: Problems on $$\mathcal{C}_{\sigma}$$ -Sets, $$\sigma$$ -Additivity, and Permutable Series


## Exercise $$\PageIndex{1}$$

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

Prove Note 3.

## Exercise $$\PageIndex{2}$$

Show that every open set $$A \neq \emptyset$$ in $$E^{n}$$ is a countable union of disjoint half-open cubes.
[Outline: For each natural $$m,$$ show that $$E^{n}$$ is split into such cubes of edge length $$2^{-m}$$ by the hyperplanes
$x_{k}=\frac{i}{2^{m}} \quad i=0, \pm 1, \pm 2, \ldots ; k=1,2, \ldots, n ,$
and that the family $$\mathcal{C}_{m}$$ of such cubes is countable.
For $$m>1,$$ let $$C_{m 1}, C_{m 2}, \ldots$$ be the sequence of those cubes from $$\mathcal{C}_{m}$$ (if any) that lie in $$A$$ but not in any cube $$C_{s j}$$ with $$s<m .$$
$$\left.\text { As } A \text { is open, } x \in A \text { iff } x \in \text { some } C_{m j} .\right]$$

## Exercise $$\PageIndex{3}$$

Prove that any open set $$A \subseteq E^{1}$$ is a countable union of disjoint (possibly infinite) open intervals.
[Hint: By Lemma $$2, A=\bigcup_{n}\left(a_{n}, b_{n}\right) .$$ If, say, $$\left(a_{1}, b_{1}\right)$$ overlaps with some $$\left(a_{m}, b_{m}\right)$$, replace both by their union. Continue inductively.

## Exercise $$\PageIndex{4}$$

Prove that $$\mathcal{C}_{\sigma}$$ is closed under finite intersections and countable unions.

## Exercise $$\PageIndex{5}$$

(i) Find $$A, B \in \mathcal{C}_{\sigma}$$ such that $$A-B \notin \mathcal{C}_{\sigma}$$
(ii) Show that $$\mathcal{C}_{\sigma}$$ is not a semiring.
[Hint: Try $$A=E^{1}, B=R$$ (the rationals).]
Note. In the following problems, $$J$$ is countably infinite, $$a_{i} \in E(E \text { complete). }$$

## Exercise $$\PageIndex{6}$$

Prove that
$\sum_{i \in J}\left|a_{i}\right|<\infty$
iff for every $$\varepsilon>0,$$ there is a finite set
$F \subset J \quad(F \neq \emptyset)$
such that
$\sum_{i \in I}\left|a_{i}\right|<\varepsilon$
for every finite $$I \subset J-F$$.
[Outline: By Theorem 2, fix $$u : N onto_{\iff} J$$ with
$\sum_{i \in J}\left|a_{i}\right|=\sum_{n=1}^{\infty}\left|a_{u_{n}}\right| .$
By Cauchy's criterion,
$\sum_{n=1}^{\infty}\left|a_{u_{n}}\right|<\infty$
iff
$(\forall \varepsilon>0)(\exists q)(\forall n>m>q) \quad \sum_{k=m}^{n}\left|a_{u_{k}}\right|<\varepsilon .$
Let $$F=\left\{u_{1}, \ldots, u_{q}\right\} .$$ If $$I$$ is as above,
$(\exists n>m>q) \quad\left\{u_{m}, \ldots, u_{n}\right\} \supseteq I ;$
so
$\left.\sum_{i \in I}\left|a_{i}\right| \leq \sum_{k=m}^{n}\left|a_{u_{k}}\right|<\varepsilon .\right]$

## Exercise $$\PageIndex{7}$$

Prove that if
$\sum_{i \in J}\left|a_{i}\right|<\infty ,$
then for every $$\varepsilon>0,$$ there is a finite $$F \subset J(F \neq \emptyset)$$ such that
$\left|\sum_{i \in J} a_{i}-\sum_{i \in K} a_{i}\right|<\varepsilon$
for each finite $$K \supset F(K \subset J)$$.
[Hint: Proceed as in Problem $$6,$$ with $$I=K-F$$ and $$q$$ so large that
$\left|\sum_{i \in J} a_{i}-\sum_{i \in F} a_{i}\right|<\frac{1}{2} \varepsilon \quad \text { and } \quad\left|\sum_{i \in F} a_{i}\right|<\frac{1}{2} \varepsilon . ]$

## Exercise $$\PageIndex{8}$$

Show that if
$J=\bigcup_{n=1}^{\infty} I_{n}(\text {disjoint}) ,$
then
$\sum_{i \in J}\left|a_{i}\right|=\sum_{n=1}^{\infty} b_{n}, \text { where } b_{n}=\sum_{i \in I_{n}}\left|a_{i}\right| .$
(Use Problem 8' below.)

## Exercise $$\PageIndex{8'}$$

Show that
$\sum_{i \in J}\left|a_{i}\right|=\sup _{F} \sum_{i \in F}\left|a_{i}\right|$
over all finite sets $$F \subset J(F \neq \emptyset)$$.
$$\text { [Hint: Argue as in Theorem } 2 .]$$

## Exercise $$\PageIndex{9}$$

Show that if $$\emptyset \neq I \subseteq J,$$ then
$\sum_{i \in I}\left|a_{i}\right| \leq \sum_{i \in J}\left|a_{i}\right| .$
$$\left.\text { [Hint: Use Problem } 8^{\prime} \text { and Corollary } 2 \text { of Chapter } 2, §§8-9 .\right]$$

## Exercise $$\PageIndex{10}$$

Continuing Problem $$8,$$ prove that if
$\sum_{i \in J}\left|a_{i}\right|=\sum_{n=1}^{\infty} b_{n}<\infty ,$
then
$\sum_{i \in J} a_{i}=\sum_{n=1}^{\infty} c_{n} \text { with } c_{n}=\sum_{i \in I_{n}} a_{i} .$
[Outline: By Problem 9,
$(\forall n) \sum_{i \in I_{n}}\left|a_{i}\right|<\infty ;$
so
$c_{n}=\sum_{i \in I_{n}} a_{i}$
and
$\sum_{n=1}^{\infty} c_{n}$
converge absolutely.
Fix $$\varepsilon$$ and $$F$$ as in Problem $$7 .$$ Choose the largest $$q \in N$$ with
$F \cap I_{q} \neq \emptyset$
(why does it exist?), and fix any $$n>q .$$ By Problem $$7,(\forall k \leq n)$$
\begin{aligned}(\forall k \leq n)\left(\exists \text { finite } F_{k} | J \supseteq F_{k} \supseteq F \cap I_{q}\right) & \\\left(\forall \text { finite } H_{k} | I_{k} \supseteq H_{k} \supseteq F_{k}\right) &\left|\sum_{i \in H_{k}} a_{i}-\sum_{k=1}^{n} c_{k}\right|<\frac{1}{2} \varepsilon \end{aligned} .
(Explain!) Let
$K=\bigcup_{k=1}^{n} H_{k} ;$
so
$\left|\sum_{k=1}^{n} c_{k}-\sum_{i \in J} a_{i}\right|<\varepsilon$
and $$K \supset F .$$ By Problem 7,
$\left|\sum_{i \in K} a_{i}-\sum_{i \in J} a_{i}\right|<\varepsilon .$
Deduce
$\left|\sum_{k=1}^{n} c_{k}-\sum_{i \in J} a_{i}\right|<2 \varepsilon .$
$$\text { Let } n \rightarrow \infty ; \text { then } \varepsilon \rightarrow 0 .]$$

## Exercise $$\PageIndex{11}$$

(Double series.) Prove that if one of the expressions
$\sum_{n, k=1}^{\infty}\left|a_{n k}\right|, \quad \sum_{n=1}^{\infty}\left(\sum_{k=1}^{\infty}\left|a_{n k}\right|\right), \quad \sum_{k=1}^{\infty}\left(\sum_{n=1}^{\infty}\left|a_{n k}\right|\right)$
is finite, so are the other two, and
$\sum_{n, k} a_{n k}=\sum_{n}\left(\sum_{k} a_{n k}\right)=\sum_{k}\left(\sum_{n} a_{n k}\right) ,$
with all series involved absolutely convergent.
[Hint: Use Problems 8 and $$10,$$ with $$J=N \times N,$$
$I_{n}=\{(n, k) \in J | k=1,2, \ldots\} \text { for each } n ;$
so
$b_{n}=\sum_{k=1}^{\infty}\left|a_{n k}\right| \text { and } c_{n}=\sum_{k=1}^{\infty} a_{n k} .$
Thus obtain
$\sum_{n, k} a_{n k}=\sum_{n} \sum_{k} a_{n k} .$
Similarly,
$\left.\sum_{n, k} a_{n k}=\sum_{k} \sum_{n} a_{n, k} .\right]$