7.2.E: Problems on \(\mathcal{C}_{\sigma}\) -Sets, \(\sigma\) -Additivity, and Permutable Series
Fill in the missing details in the proofs of this section.
Prove Note 3.
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 \(s<m .\)
\(\left.\text { As } A \text { is open, } x \in A \text { iff } x \in \text { some } C_{m j} .\right]\)
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.
Prove that \(\mathcal{C}_{\sigma}\) is closed under finite intersections and countable unions.
(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, \(a_{i} \in E(E \text { complete). }\)
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
\[
\left.\sum_{i \in I}\left|a_{i}\right| \leq \sum_{k=m}^{n}\left|a_{u_{k}}\right|<\varepsilon .\right]
\]
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
\[
\left|\sum_{i \in J} a_{i}-\sum_{i \in F} a_{i}\right|<\frac{1}{2} \varepsilon \quad \text { and } \quad\left|\sum_{i \in F} a_{i}\right|<\frac{1}{2} \varepsilon . ]
\]
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.)
Show that
\[
\sum_{i \in J}\left|a_{i}\right|=\sup _{F} \sum_{i \in F}\left|a_{i}\right|
\]
over all finite sets \(F \subset J(F \neq \emptyset)\).
\(\text { [Hint: Argue as in Theorem } 2 .]\)
Show that if \(\emptyset \neq I \subseteq J,\) then
\[
\sum_{i \in I}\left|a_{i}\right| \leq \sum_{i \in J}\left|a_{i}\right| .
\]
\(\left.\text { [Hint: Use Problem } 8^{\prime} \text { and Corollary } 2 \text { of Chapter } 2, §§8-9 .\right]\)
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
\[
\left|\sum_{k=1}^{n} c_{k}-\sum_{i \in J} a_{i}\right|<2 \varepsilon .
\]
\(\text { Let } n \rightarrow \infty ; \text { then } \varepsilon \rightarrow 0 .]\)
(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,
\[
\left.\sum_{n, k} a_{n k}=\sum_{k} \sum_{n} a_{n, k} .\right]
\]