Corollary

— Suppose that $\mu$ is the sum of a family $(\mu_\alpha)_{\alpha \in A}$ of measures on T. For a positive numerical function g defined on T to be locally $\mu$-integrable, it is necessary and sufficient that g be locally $\mu_\alpha$-integrable for every $\alpha \in A$ and that the family $(g \cdot \mu_\alpha)_{\alpha \in A}$ be summable. In this case,

Let $(g_\alpha)_{\alpha \in A}$ be a family of $\mu$-measurable positive functions defined on T. Let $S_\alpha$ be the set of $t \in T$ such that $g_\alpha(t) \neq 0$. We shall say that the family $(g_\alpha)$ is locally countable if the family $(S_\alpha)$ is locally countable (Ch. IV, §5, No. 9); this amounts to saying that, for every compact set K in T, the set of $\alpha \in A$ such that $g_\alpha|K$ is not zero is countable.