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Mathematics LibreTexts

8.12.E: Problems on Differentiation and Related Topics

( \newcommand{\kernel}{\mathrm{null}\,}\)

Exercise 8.12.E.1

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

Exercise 8.12.E.2

Given a measure s:ME(M¯K), prove that
(i) s is topological;
(ii) its Borel restriction σ is strongly regular; and
(iii) D_s,¯Ds, and s do not change if s or m are restricted to the Borel field B in En; neither does this affect the propositions on ¯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 ¯K.]

Exercise 8.12.E.3

What analogues to 2(i)( iii ) apply to Ω-differentiation in En?In(S,ρ)?

Exercise 8.12.E.4

(i) Show that any m-singular measure s in En, finite on ¯K, has a zero derivative (a.e.).
(ii) For Ω-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 σ).
Suppose
mEn(¯Ds>0)>a>0


and find a contradiction to Lemma 2.]

Exercise 8.12.E.5

Give another proof for Theorem 4 in Chapter 7,812.
[Hint: Fix an open cube J¯K. By Problem 2(iii), restrict s and m to
M0={XB|XJ}


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 Ω-differentiation, assume s regular; argue as in Corollary 1, using Corollary 2
of 11.]

Exercise 8.12.E.6

Prove that if
F(x)=Lxafdm(axb),


with f:E1E(En,Cn)m-integrable on A=[a,b], then F is differentiable, with F=f, a.e. on A.
[Hint: Via components, reduce all to the case f0,F on A.
Let
s=fdm

on M. Let t=mF be the F-induced LS measure. Show that s=t on intervals in A; so s=t=F 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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