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Mathematics LibreTexts

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

( \newcommand{\kernel}{\mathrm{null}\,}\)

Exercise 7.2.E.1

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

Exercise 7.2.E.1

Prove Note 3.

Exercise 7.2.E.2

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 2m 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, xA iff x some Cmj.]

Exercise 7.2.E.3

Prove that any open set AE1 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.

Exercise 7.2.E.4

Prove that Cσ is closed under finite intersections and countable unions.

Exercise 7.2.E.5

(i) Find A,BCσ such that ABCσ
(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, aiE(E complete). 

Exercise 7.2.E.6

Prove that
iJ|ai|<
iff for every ε>0, there is a finite set
FJ(F)
such that
iI|ai|<ε
for every finite IJF.
[Outline: By Theorem 2, fix u:NontoJ with
iJ|ai|=n=1|aun|.
By Cauchy's criterion,
n=1|aun|<
iff
(ε>0)(q)(n>m>q)nk=m|auk|<ε.
Let F={u1,,uq}. If I is as above,
(n>m>q){um,,un}I;
so
iI|ai|nk=m|auk|<ε.]

Exercise 7.2.E.7

Prove that if
iJ|ai|<,
then for every ε>0, there is a finite FJ(F) such that
|iJaiiKai|<ε
for each finite KF(KJ).
[Hint: Proceed as in Problem 6, with I=KF and q so large that
|iJaiiFai|<12ε and |iFai|<12ε.]

Exercise 7.2.E.8

Show that if
J=n=1In(disjoint),
then
iJ|ai|=n=1bn, where bn=iIn|ai|.
(Use Problem 8' below.)

Exercise 7.2.E.8

Show that
iJ|ai|=supFiF|ai|
over all finite sets FJ(F).
 [Hint: Argue as in Theorem 2.]

Exercise 7.2.E.9

Show that if IJ, then
iI|ai|iJ|ai|.
\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]


7.2.E: Problems on \mathcal{C}_{\sigma} -Sets, \sigma -Additivity, and Permutable Series is shared under a CC BY 1.0 license and was authored, remixed, and/or curated by LibreTexts.

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