Exercise 4, § 1

Let T, X, Y be three locally compact spaces, with X being countable at infinity. Let $\mu$ be a positive measure on T, $(\lambda_t)_{t \in T}$ a $\mu$-adequate$^1$ family of positive measures on X, and $\nu = \int \lambda_t d\mu(t)$. Assume that one of the following two conditions holds: a) Y has a countable base; b) the measure $\nu$ is bounded. Under these conditions, if $\pi$ is a $\nu$-proper mapping of X into Y, then the set of $t \in T$ such that $\pi$ is not $\lambda_t$-proper is locally $\mu$-negligible; moreover, the family $(\pi(\lambda_t))_{t \in T}$ of positive measures on Y (defined locally almost everywhere in T) is $\mu$-adequate, and

(In case a), apply Exer. 2 to $\rho_x = \varepsilon_{\pi(x)}$. In case b), note that $\lambda_t$ is bounded locally almost everywhere, and reduce to showing that the mapping $t \mapsto \pi(\lambda_t)$ is vaguely $\mu$-measurable (cf. Ch. V, §3, No. 2, Prop. 4). Given a number $\varepsilon > 0$ and a compact subset K of T, observe first that there exists in X an increasing sequence $(F_m)$ of compact sets such that the restriction of $\pi$ to each $F_m$ is continuous and such that, on setting $N_m = X - F_m$, $\nu(N_m) \leq \varepsilon/4^m$. Let $A_m$ be the set of $t \in K$ such that $N_m$ is not $\lambda_t$-integrable or such that $\lambda_t(N_m) > 1/2^m$; if A is the union of the $A_m$, $\mu(A) \leq \varepsilon/2$. Let $K_1$ be a compact subset of $K - A$ such that $\mu(K - K_1) \leq \varepsilon$ and such that, for every function $g \in \mathscr{K}(X)$, the restriction of $t \mapsto \int g(x) d\lambda_t(x)$ to $K_1$ is continuous. Using Urysohn’s theorem, show that for every function $f \in \mathscr{K}(Y)$, the restriction to $K_1$ of $t \mapsto \int f(\pi(x)) d\lambda_t(x)$ is the uniform limit of continuous functions.)