8.12.E: Problems on Differentiation and Related Topics
Fill in all proof details in this section. Verify footnote 4 and Note 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 \(\overline{\mathcal{K} .]}\)
What analogues to \(2(\mathrm{i})-(\text { iii })\) apply to \(\Omega\)-differentiation in \(E^{n} ? \operatorname{In}(S, \rho) ?\)
(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 \(\sigma\)).
Suppose
\[
m E^{n}(\overline{D} s>0)>a>0
\]
and find a contradiction to Lemma 2.]
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.]
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 \(A .\)
Let
\[
s=\int f d m
\]
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.]