Proposition 3

For a set $A$ to be measurable, it is necessary and sufficient that, for every compact set $K$, $A \cap K$ be integrable.

The condition is necessary, because the union of a sequence of integrable sets $A_n$ is integrable when $\sum_n |\mu|(A_n)$ is finite (§ 4, No. 5, Cor. of Prop. 8). The condition is sufficient because, for every integrable set $B$, there exist a negligible set $N \subset B$ and a partition of $B - N$ into a sequence of compact sets (§ 4, No. 6, Cor. 2 of Th. 4).