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Mathematics LibreTexts

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

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    32343
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    Exercise \(\PageIndex{1}\)

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

    Exercise \(\PageIndex{1'}\)

    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,
    \[
    \left.\sum_{n, k} a_{n k}=\sum_{k} \sum_{n} a_{n, k} .\right]
    \]

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