Theorem 2

— Let T be a locally compact space, or a Polish space, and let H be a relatively compact subset of $\mathscr{M}_+^b(T)$; then H satisfies Prokhorov’s condition.

We may restrict ourselves to the case that H is closed, hence compact. The total masses of the measures $\mu \in H$ are obviously bounded, because the mapping $\mu \mapsto \mu(1)$ is continuous, and everything comes down to proving the assertion b) of Def. 2.

Suppose first that T is locally compact. Let $\varepsilon$ be a number > 0. Let us associate to every measure $\mu \in H$ a compact set $K_\mu$ in T such that $\mu^\bullet(T - K_\varepsilon) < \varepsilon$, then a relatively compact open neighborhood $U_\mu$ of $K_\mu$. The function $\lambda \mapsto \lambda^\bullet(T - U_\mu)$ being upper semi-continuous on $\mathscr{M}_+^b(T)$ (No. 3, Prop. 6), the set $V^\mu$ of measures $\lambda \in H$ such that $\lambda^\bullet(T - U_\mu) < \varepsilon$ is a neighborhood of $\mu$ in H. Therefore there exists a finite subset $H'$ of H such that the sets $V^\mu$ ($\mu \in H'$) cover H. Denoting by K the compact set $\bigcup_{\mu \in H'} \overline{U}_\mu$, we have $\lambda^\bullet(T - K) < \varepsilon$ for all $\lambda \in H$.

Suppose next that T is Polish. We do not restrict the generality by assuming that T is the intersection of a decreasing sequence $(T_p)_{p \geq 1}$ of open subsets of a compact space X (GT, IX, §6, No. 1, Cor. 1 of Th. 1). Let $i_p$ be the injection of T into $T_p$, and let $H_p$ be the set of measures of the form $i_p(\lambda)$ for $\lambda \in H$; since $H_p$ is compact in $\mathscr{M}_+^b(T_p)$, it follows that there exists a compact set $K_p \subset T_p$ such that $\nu^\bullet(T_p - K_p) \leq \varepsilon 2^{-p}$ for every measure $\nu \in H_p$, by the preceding result applied to the locally compact space $T_p$. Therefore also $\nu^\bullet(T - (T \cap K_p)) \leq \varepsilon 2^{-p}$, and finally $\lambda^\bullet(T - (T \cap K_p)) \leq \varepsilon 2^{-p}$ for every measure $\lambda \in H$. Now set $K = \bigcap_p K_p$; the set K is compact and is contained in T, and, for every measure $\lambda \in H$, we have $\lambda^\bullet(T - K) \leq \sum_p \lambda^\bullet(T - (T \cap K_p)) \leq \sum_p \varepsilon 2^{-p} = \varepsilon$. Prokhorov’s condition is thus verified.