7.2.E: Problems on Cσ -Sets, σ -Additivity, and Permutable Series
( \newcommand{\kernel}{\mathrm{null}\,}\)
Fill in the missing details in the proofs of this section.
Prove Note 3.
Show that every open set A≠∅ in En is a countable union of disjoint half-open cubes.
[Outline: For each natural m, show that En is split into such cubes of edge length 2−m by the hyperplanes
xk=i2mi=0,±1,±2,…;k=1,2,…,n,
and that the family Cm of such cubes is countable.
For m>1, let Cm1,Cm2,… be the sequence of those cubes from Cm (if any) that lie in A but not in any cube Csj with s<m.
As A is open, x∈A iff x∈ some Cmj.]
Prove that any open set A⊆E1 is a countable union of disjoint (possibly infinite) open intervals.
[Hint: By Lemma 2,A=⋃n(an,bn). If, say, (a1,b1) overlaps with some (am,bm), replace both by their union. Continue inductively.
Prove that Cσ is closed under finite intersections and countable unions.
(i) Find A,B∈Cσ such that A−B∉Cσ
(ii) Show that Cσ is not a semiring.
[Hint: Try A=E1,B=R (the rationals).]
Note. In the following problems, J is countably infinite, ai∈E(E complete).
Prove that
∑i∈J|ai|<∞
iff for every ε>0, there is a finite set
F⊂J(F≠∅)
such that
∑i∈I|ai|<ε
for every finite I⊂J−F.
[Outline: By Theorem 2, fix u:Nonto⟺J with
∑i∈J|ai|=∞∑n=1|aun|.
By Cauchy's criterion,
∞∑n=1|aun|<∞
iff
(∀ε>0)(∃q)(∀n>m>q)n∑k=m|auk|<ε.
Let F={u1,…,uq}. If I is as above,
(∃n>m>q){um,…,un}⊇I;
so
∑i∈I|ai|≤n∑k=m|auk|<ε.]
Prove that if
∑i∈J|ai|<∞,
then for every ε>0, there is a finite F⊂J(F≠∅) such that
|∑i∈Jai−∑i∈Kai|<ε
for each finite K⊃F(K⊂J).
[Hint: Proceed as in Problem 6, with I=K−F and q so large that
|∑i∈Jai−∑i∈Fai|<12ε and |∑i∈Fai|<12ε.]
Show that if
J=∞⋃n=1In(disjoint),
then
∑i∈J|ai|=∞∑n=1bn, where bn=∑i∈In|ai|.
(Use Problem 8' below.)
Show that
∑i∈J|ai|=supF∑i∈F|ai|
over all finite sets F⊂J(F≠∅).
[Hint: Argue as in Theorem 2.]
Show that if ∅≠I⊆J, then
∑i∈I|ai|≤∑i∈J|ai|.
\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]