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8.12.E: Problems on Differentiation and Related Topics

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  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    Exercise \(\PageIndex{1}\)

    Fill in all proof details in this section. Verify footnote 4 and Note 2.

    Exercise \(\PageIndex{2}\)

    Given a measure \(s: \mathcal{M}^{\prime} \rightarrow E^{*}\left(\mathcal{M}^{\prime} \supseteq \overline{\mathcal{K}}\right),\) prove that
    (i) \(s\) is topological;
    (ii) its Borel restriction \(\sigma\) is strongly regular; and
    (iii) \(\underline{D} s, \overline{D} s,\) and \(s^{\prime}\) do not change if \(s\) or \(m\) are restricted to the Borel field \(\mathcal{B}\) in \(E^{n} ;\) neither does this affect the propositions on \(\overline{\mathcal{K}}\)-differentiation proved here.
    [Hints: (i) Use Lemma 2 of Chapter 7, §2. (ii) Use also Problem 10 in Chapter 7, §7. (iii) All depends on \(\overline{\mathcal{K} .]}\)

    Exercise \(\PageIndex{3}\)

    What analogues to \(2(\mathrm{i})-(\text { iii })\) apply to \(\Omega\)-differentiation in \(E^{n} ? \operatorname{In}(S, \rho) ?\)

    Exercise \(\PageIndex{4}\)

    (i) Show that any \(m\)-singular measure \(s\) in \(E^{n},\) finite on \(\overline{\mathcal{K}},\) has a zero derivative (a.e.).
    (ii) For \(\Omega\)-derivatives, prove that this holds if \(s\) is also regular.
    [Hint for (i): By Problem 2, we may assume s regular (if not, replace it by \(\sigma\)).
    m E^{n}(\overline{D} s>0)>a>0
    and find a contradiction to Lemma 2.]

    Exercise \(\PageIndex{5}\)

    Give another proof for Theorem 4 in Chapter \(7,812 .\)
    [Hint: Fix an open cube \(J \in \overline{\mathcal{K}} .\) By Problem 2(iii), restrict \(s\) and \(m\) to
    \mathcal{M}_{0}=\{X \in \mathcal{B} | X \subseteq J\}
    to make them finite. Apply Corollary 2 in §11 to \(s\). Then use Problem \(4,\) Theorem 1 of the present section, and Theorem 1 of Chapter 7, §12.
    For \(\Omega\)-differentiation, assume \(s\) regular; argue as in Corollary \(1,\) using Corollary 2
    of 11.]

    Exercise \(\PageIndex{6}\)

    Prove that if
    F(x)=L \int_{a}^{x} f d m \quad(a \leq x \leq b) ,
    with \(f: E^{1} \rightarrow E^{*}\left(E^{n}, C^{n}\right) m\)-integrable on \(A=[a, b],\) then \(F\) is differentiable, with \(F^{\prime}=f,\) a.e. on \(A .\)
    [Hint: Via components, reduce all to the case \(f \geq 0, F \uparrow\) on \(A .\)
    s=\int f d m
    on \(\mathcal{M}^{*}\). Let \(t=m_{F}\) be the \(F\)-induced LS measure. Show that \(s=t\) on intervals in \(A\); so \(s^{\prime}=t^{\prime}=F^{\prime}\) a.e. on \(A\) (Problem 9 in Chapter 7, §11). Use Theorem 1.]

    8.12.E: Problems on Differentiation and Related Topics is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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