Exercise 10, § 5

let T be a completely regular space; denote by $\mathscr{S}$ the set of real functions $f \geq 1$ on T, such that the set of points t of T for which $f(t) \leq c$ is compact for every real number c. For every $f \in \mathscr{S}$, denote by $\mathscr{M}_f$ the set of bounded measures $\mu$ on T such that $|\mu|^\bullet(f) \leq 1$.

a) For a subset H of $\mathscr{M}^b(T)$ to satisfy Prokhorov’s condition, it is necessary and sufficient that there exist an $f \in \mathscr{S}$ such that $H \subset \mathscr{M}_f$ (for necessity, take $f$ to be of the form $c(1 + \sum_{n \geq 1} n \cdot f_n)$ where $f_n$ is the characteristic function of a set $U_n$ such that $T - U_n$ is compact and $\sup_{\mu \in H} |\mu|^\bullet(U_n) \leq 2^{-n}$).

b) Suppose T is locally compact. For a subset H of $\mathscr{M}^b(T)$ to be relatively compact (for the tight topology), it is necessary and sufficient that there exist a continuous function $f \in \mathscr{S}$ such that $H \subset \mathscr{M}_f$.

c) Let C be a closed convex cone in $\mathscr{M}_+^b(T)$. Show that $C \cap \mathscr{M}_f$ is a cap of C (TVS, II, §7, No. 2, Def. 3) for every $f \in \mathscr{S}$, and deduce from this that C is the union of its caps. Denote by E the union of the extremal generators of C. Suppose T is Souslin. Show that for every $\pi \in C$, there exist a Borel subset B of $\mathscr{M}_+^b(T)$ contained in E and a positive measure P on B of total mass 1, such that $\pi = \int_B \mu dP(\mu)$ (apply Choquet’s integral representation theorem).