Proposition 7

— Let X be a locally compact space that is countable at infinity, and let $t \mapsto \lambda_t$ be a $\mu$-adequate mapping of T into $\mathscr{M}_+(X)$.

Let $g$ be a positive numerical function defined on $X$, locally integrable for the measure $\nu = \int \lambda_t d\mu(t)$. The set of $t \in T$ such that $g$ is not locally $\lambda_t$-integrable is then locally negligible for $\mu$, the mapping $t \mapsto g \cdot \lambda_t$ (defined locally $\mu$-almost everywhere) is $\mu$-adequate, and

Let $(K_n)_{n \in \mathbf{N}}$ be an increasing sequence of compact subsets of $X$ whose interiors cover $X$; if $\eta$ is any positive measure on $X$, to say that $g$ is locally $\eta$-integrable is equivalent to saying that $g \varphi_{K_n}$ is $\eta$-integrable for every $n$. Now let $H_n$ be the set of $t \in T$ such that $g \varphi_{K_n}$ is not $\lambda_t$-integrable, and let $H = \bigcup_n H_n$; since $H_n$ is locally $\mu$-negligible for all $n$ (§ 3, No. 3, Th. 1), the same is true of $H$, which establishes the first assertion of the statement. Replacing $\lambda_t$ by 0 for $t$ in $H$ (which does not change the measure $\nu$), we can suppose that $g$ is locally $\lambda_t$-integrable for every $t \in T$. For every $\nu$-measurable positive function $h$ defined on $X$, we have, by Prop. 3 and by Prop. 5 of § 3, No. 2,

This formula and Prop. 5 of § 3, No. 2 first show (on taking $h \in \mathscr{K}_+(X)$) that the mapping $t \mapsto g \cdot \lambda_t$ is scalarly essentially $\mu$-integrable, and that its integral is $g \cdot \nu$; in other words, the relation (9) holds. Next, let us replace $\mu$ by a positive measure $\mu' \leq \mu$, and let us take for $h$ a positive lower semi-continuous function: it follows at once from these relations that $t \mapsto g \cdot \lambda_t$ is $\mu$-adequate (§ 3, No. 1, Def. 1).