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Mathematics LibreTexts

8.5.E: Problems on Integration of Extended-Real Functions

  • Page ID
    32372
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    Exercise \(\PageIndex{1}\)

    Using the formulas in ( 1) and our conventions, verify that
    (i) \(\overline{\int}_{A} f=+\infty\) iff \(\overline{\int}_{A} f^{+}=\infty\);
    (ii) \(\underline{\int}_{A} f=\infty\) iff \(\underline{\int}_{A} f^{+}=\infty ;\) and
    (iii) \(\overline{f}_{A} f=-\infty\) iff \(\underline{\int}_{A} f^{-}=\infty\) and \(\overline{\int}_{A} f^{+}<\infty\).
    (iv) Derive a condition similar to (iii) for \(\underline{\int}_{A} f=-\infty\).
    (v) Review Problem 6 of Chapter 4, §4.

    Exercise \(\PageIndex{2}\)

    Fill in the missing proof details in Theorems 1 to 3 and Lemmas 1 and 2.

    Exercise \(\PageIndex{3}\)

    Prove that if \(\underline{\int_{A}} f=\infty,\) there is an elementary and (extended) real map \(g \leq f\) on \(A,\) with \(\int_{A} g=\infty\).
    [Outline: By Problem \(1,\) we have
    \[
    \underline{\int_{A}} f^{+}=\infty .
    \]
    As Lemmas 1 and 2 surely hold for nonnegative functions, fix a measurable \(F \leq f^{+}\) \((F \geq 0),\) with
    \[
    \int_{A} F=\underline{\int_{A}} f^{+}=\infty .
    \]
    Arguing as in Theorem \(3,\) find an elementary and nonnegative map \(g \leq F,\) with
    \[
    (1+\varepsilon) \int_{A} g=\int_{A} F=\infty ;
    \]
    so \(\int_{A} g=\infty\) and \(0 \leq g \leq F \leq f^{+}\) on \(A\).
    Let
    \[
    A_{+}=A(F>0) \in \mathcal{M}
    \]
    and
    \[
    A_{0}=A(F=0) \in \mathcal{M}
    \]
    (Theorem 1 in §2). On \(A_{+},\)
    \[
    g \leq F \leq f^{+}=f(\text { why? }) ,
    \]
    while on \(A_{0}, g=F=0 ;\) so
    \[
    \int_{A_{+}} g=\int_{A} g=\infty(\mathrm{why} ?) .
    \]
    Now redefine \(g=-\infty\) on \(A_{0}\) (only). Show that \(g\) is then the required function.]

    Exercise \(\PageIndex{4}\)

    For any \(f: S \rightarrow E^{*},\) prove the following.
    (a) If \(\overline{\int}_{A} f<\infty,\) then \(f<\infty\) a.e. on \(A\).
    (b) If \(\underline{\int_{A}} f\) is orthodox and \(>-\infty,\) then \(f>-\infty\) a.e. on \(A\).
    [Hint: Use Problem 1 and apply Corollary 1 to \(f^{+} ;\) thus prove (a). Then for (b), use Theorem 1(e').]

    Exercise \(\PageIndex{5}\)

    \(\Rightarrow 5\). For any \(f, g: S \rightarrow E^{*},\) prove that
    (i) \(\overline{\int}_{A} f+\overline{\int}_{A} g \geq \overline{\int}_{A}(f+g),\) and
    (ii) \(\underline{\int}_{A}(f+g) \geq \underline{\int}_{A} f+\underline{\int}_{A} g \quad\) if \(\left|\underline{\int}_{A} g\right|<\infty\).
    [Hint: Suppose that
    \[
    \overline{\int}_{A} f+\overline{\int}_{A} g<\overline{\int}_{A}(f+g) .
    \]
    Then there are numbers
    \[
    u>\overline{\int}_{A} f \text { and } v>\overline{\int}_{A} g ,
    \]
    with
    \[
    u+v \leq
    overline{\int}_{A}(f+g) .
    \]
    (Why?) Thus Lemma 1 yields elementary and (extended) real maps \(F \geq f\) and \(G \geq g\) such that
    \[
    u>\overline{\int}_{A} F \text { and } v>\overline{\int}_{A} G
    \]
    As \(f+g \leq F+G\) on \(A,\) Theorem \(1(\mathrm{c})\) of §5 and Problem 6 of §4 show that
    \[
    \overline{\int}_{A}(f+g) \leq \int_{A}(F+G)=\int_{A} F+\int_{A} G<u+v ,
    \]
    contrary to
    \[
    u+v \leq \overline{\int}_{A}(f+g) .
    \]
    Similarly prove clause (ii).]

    Exercise \(\PageIndex{6}\)

    Continuing Problem \(5,\) prove that
    \[
    \overline{\int}_{A}(f+g) \geq \overline{\int}_{A} f+\underline{\int}_{A} g \geq \underline{\int}_{A}(f+g) \geq \underline{\int}_{A} f+\underline{\int}_{A} g
    \]
    provided \(\left|\underline{\int}_{A} g\right|<\infty\).
    [Hint for the second inequality: We may assume that
    \[
    \overline{\int}_{A}(f+g)<\infty \text { and } \overline{\int}_{A} f>-\infty .
    \]
    (Why?) Apply Problems 5 and \(4(\mathrm{a})\) to
    \[
    \overline{\int}_{A}((f+g)+(-g)) .
    \]
    Use Theorem \(\left.1\left(\mathrm{e}^{\prime}\right) .\right]\)

    Exercise \(\PageIndex{7}\)

    Prove the following.
    (i) \[
    \overline{\int}_{A}|f|<\infty \text { iff }-\infty<\underline{\int}_{A} f \leq \overline{\int}_{A} f<\infty .
    \]
    (ii) If \(\overline{f}_{A}|f|<\infty\) and \(\overline{\int}_{A}|g|<\infty,\) then
    \[
    \left|\overline{\int}_{A} f-\overline{\int}_{A} g\right| \leq \overline{\int}_{A}|f-g|
    \]
    and
    \[
    \left|\underline{\int}_{A} f-\underline{\int}_{A} g\right| \leq \overline{\int}_{A}|f-g| .
    \]
    [Hint: Use Problems \(5 \text { and } 6 .]\)

    Exercise \(\PageIndex{8}\)

    Show that any signed measure \(\left.\bar{s}_{f} \text { (Note } 4\right)\) is the difference of two measures: \(\bar{s}_{f}=\bar{s}_{f+}-\bar{s}_{f-}\).

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