Exercise 3, § 5

Let T be a Polish space and $(\mu_n)$ a sequence of bounded positive measures on T converging tightly to a measure $\mu$. Denote by $\mathscr{C}$ the set of subsets A of T having the following property: for every $\varepsilon > 0$, there exists a compact subset K of A such that $\sup_{n} \mu_n(A - K) < \varepsilon$. Let $\mathscr{D}$ be the set of subsets A of T that, together with their complement, belong to $\mathscr{C}$.

a) If the sets A and $A'$ belong to $\mathscr{C}$, then so do $A \cup A'$ and $A \cap A'$; deduce from this that $\mathscr{D}$ is a clan of subsets of T.

b) Every subset A of T whose boundary is $\mu$-negligible belongs to $\mathscr{D}$ (apply Prokhorov’s theorem to the interior of A, and use Exer. 2 e)).

c) Every $A \in \mathscr{D}$ is a convergence set for the sequence $(\mu_n)$ (cf. Exer. 2); conversely, every open or closed convergence set belongs to $\mathscr{D}$.

d) Let $A \in \mathscr{D}$. Show that there exist a sequence of disjoint compact sets $K_p$ and a subset N of T having the following properties: $\alpha)$ each $K_p$ is a convergence set for the sequence $(\mu_n)$; $\beta)$ $A \subset N \cup \bigcup_p K_p$ and $\bigcup_p K_p \subset A \cup N$; $\gamma)$ for every $\varepsilon > 0$, there exists an open neighborhood U of N such that $\sup_{n} \mu_n(U) < \varepsilon$ (apply Exer. 2 f) and Prokhorov’s theorem to a suitable subspace of T that is the intersection of a sequence of open sets).