 # 12. The following result shows that under appropriate conditions, ...

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12. The following result shows that under appropriate conditions, a Borel set can be approximated from below by a compact set, and from above by an open set. Problems of this type will be examined systematically in Chapter 4. If u is a a-finite measure on 9(R"), show that for each B E 9(R"), (a) HC (B) = sup{(K): K C B, K compact). If u is in fact a Lebesgue-Stjeltjes measure, show that (b) p(B) = inf{(V): V 0 B, open}. (c) Give an example of a a-finite measure on 98 (R") that is not a Lebes- gue-Stieltjes measure and for which (b) fails.7. (Fubini's differentiation theorem) Let be increasing functions from R to R. and assume that for each x, E fn(x) converges to a finite number f(x). Show that Engl fn'(x) = f'(x) almost everywhere (Lebesgue measure). Outline: (a) It suffices to restrict the domain of all functions to [0, 1] and to as- sume all functions nonnegative. Use Fatou's lemma to show that Cerl fn'(x) < f'(x) a.e.; hence fn'(x) 0 a.e. (b) Choose ni. that Ejpne f(1)=2-k k = 1,2, and ap- ply part (a) to the functions gt (x) = f(x) - = Eliza fj(x).

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