Proposition 3

— Let $X$ be a locally compact space, $t \mapsto \lambda_t$ a $\mu$-adequate$^3$ family (Ch. V, §3, No. 1, Def. 1) of positive measures on $X$, and let $\nu = \int \lambda_t d\mu(t)$. Let $f$ be a mapping of $X$ into $F$; assume that $1^\circ$ $f$ is scalarly $\nu$-integrable; $2^\circ$ there exists a locally $\mu$-negligible set $N \subset T$ such that, for every $t \notin N$, $f$ is scalarly $\lambda_t$-integrable and $\int f d\lambda_t \in F$. Then, the function $t \mapsto \int f d\lambda_t$, defined for $t \notin N$, is scalarly essentially $\mu$-integrable, and$^4$