Exercise 13, § 5

Let $\mu$ be a positive measure on $X$. If $A$ is measurable, show that for every subset $B$ of $X$,

(if $\mu^*(B) < +\infty$, consider an integrable set $B_1$ such that $B \subset B_1$ and $\mu^*(B) = \mu(B_1)$ ($§ 4$, Exer. 7 b))). Conversely, show that if $A$ satisfies this condition, then $A$ is measurable (cf. $§ 4$, Exer. 6 e)).