Lemma 2.2.5D0A959AF6D25

Proof. Write $X=\bigcup _ mX_ m$ with measurable sets $X_ m$ of finite measure. Every member $E$ of the family satisfies $\mu (E\cap X_ m)\geq 1/k$ for some $m$ and some integer $k\geq 1$. For fixed $m$ and $k$, at most $k\mu (X_ m)$ members have this property, because the corresponding sets $E\cap X_ m$ are pairwise disjoint subsets of $X_ m$. Hence the family is a countable union of finite subfamilies. $\square$