8.12.E: Problems on Differentiation and Related Topics
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Fill in all proof details in this section. Verify footnote 4 and Note 2.
Given a measure s:M′→E∗(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.]
What analogues to 2(i)−( iii ) apply to Ω-differentiation in En?In(S,ρ)?
(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
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={X∈B|X⊆J}
Prove that if
F(x)=L∫xafdm(a≤x≤b),