# 8.12.E: Problems on Differentiation and Related Topics

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## 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 $$\sigma$$).
Suppose
$m E^{n}(\overline{D} s>0)>a>0$
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 $$A .$$
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.]

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