Proposition 13

*Let $X$ be a topological space, $\nu$ a moderated measure on $X$, $p$ a $\nu$-proper mapping of $X$ into a topological space $T$, and $\mu = p(\nu)$. Assume that every compact subspace of $X$ is metrizable. Then there exists a mapping $t \mapsto \lambda_t$ of $T$ into $\mathscr{M}_+(X)$ having the following properties:

a) for every $t \in T$, the measure $\lambda_t$ is carried by $\overline{p}^{-1}(t)$; b) for every universally measurable$^{(1)}$ positive function $f$ on $X$, the function $t \mapsto \lambda_t^\bullet(f)$ is universally measurable on $T$ and*

(1) A mapping of a topological space $X$ into a topological space $Y$ is said to be universally measurable if it is $\mu$-measurable for every measure $\mu$ on $X$ (cf. Ch. V, $§ 3$, No. 4).

c) the set of $t \in T$ such that $\lambda_t(1) \neq 1$ is locally $\mu$-negligible. Moreover, if $t \mapsto \lambda'_t$ is a mapping of $T$ into $\mathscr{M}_+(X)$ satisfying the conditions a) and b), the set of $t \in T$ such that $\lambda_t \neq \lambda'_t$ is locally $\mu$-negligible. We will need an auxiliary result: