Exercise 14, § 5

Integration, IV, p. 129

Let $(A_n)$ be a sequence of subsets of $X$ such that, for every index $n$ , there exists a measurable set $B_n \supset A_n$ , the $B_n$ being pairwise disjoint. Show that

\mu^*\left( \bigcup_n A_n \right) = \sum_n \mu^*(A_n) \quad \text{and} \quad \mu_*\left( \bigcup_n A_n \right) = \sum_n \mu_*(A_n).

Solution

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