Proposition 10

Let $T$ be a Polish space; the space $\mathscr{M}_+^b(T)$ is then Polish for the tight topology.

We begin by treating the case that $T$ is Polish and compact. The set $U$ of positive measures with mass $\leq 1$ is then compact (Ch. III, §1, No. 9, Cor. 2 of Prop. 15), and the topology induced on $U$ by the tight topology (which here coincides with the vague topology) is also induced by the topology of pointwise convergence on a total subset of $\mathscr{C}(T)$ (loc. cit., No. 10, Prop. 17). Now, there exists in $\mathscr{C}(T)$ a countable total set (GT, X, §3, No. 3, Th. 1); consequently, $U$ is a metrizable compact space. The set $V$ of positive measures of mass $< 1$ is open in $U$, hence is a Polish locally compact space. Now, the mapping $\mu \mapsto \frac{1}{1 + \mu(1)} \mu$ of $\mathscr{M}_+^b(T)$ onto $V$ is a homeomorphism, the mapping $\lambda \mapsto \frac{1}{1 - \lambda(1)} \lambda$ being the inverse homeomorphism.

Let us pass to the case that $T$ is Polish; we can suppose that $T$ is the intersection of a decreasing sequence $(G_n)$ of open sets in a metrizable compact space $X$ (GT, IX, §6, No. 1, Cor. 1 of Th. 1); the space $\mathscr{M}_+^b(T)$ is then homeomorphic to the subspace $W$ of $\mathscr{M}_+^b(X)$ consisting of the measures concentrated on $T$ (No. 3, Prop. 8), and it will suffice to show that $W$ is the intersection of a sequence of open sets in the Polish space $\mathscr{M}_+^b(X)$ (GT, loc. cit., Th. 1). Now, let $W_n$ be the set of measures $\mu \in \mathscr{M}_+^b(X)$ concentrated on $G_n$; the mapping $h_n : \mu \mapsto \mu^*(X - G_n)$ on $\mathscr{M}_+^b(X)$ is upper semi-continuous (No. 3, Prop. 6), and the set $A_k^n$ of measures $\mu \in \mathscr{M}_+^b(X)$ such that $h_n(\mu) < 1/k$ is therefore open for every $k \geq 1$ and every $n \in \mathbf{N}$. The proof is completed by observing that $W = \bigcap_n W_n = \bigcap_{n,k} A_k^n$.