Exercise 15, § 4

Let $\mu$ be a bounded positive measure on a locally compact space $X$, and let $(A_n)$ be a sequence of integrable sets in $X$ such that $\inf_n \mu(A_n) = m > 0$. Show that the set $B$ of points of $X$ that belong to infinitely many of the sets $A_n$ is integrable, and that $\mu(B) \geqslant m$.