Exercise 2, § 5

Let $T$ be a completely regular space and $\mathfrak{F}$ a filter on $\mathscr{M}_+^b(T)$ that converges tightly to a bounded measure $\mu$. A Borel subset $A$ of $T$ is said to be a convergence set (for $\mathfrak{F}$) if $\lim_{\lambda, \mathfrak{F}} \lambda_A = \mu_A$.

a) If the disjoint sets $A_1$ and $A_2$ are convergence sets, then so is $A_1 \cup A_2$.

b) Let $A$ be an open or closed set. For $A$ to be a convergence set, it is necessary and sufficient that $\lim_{\lambda, \mathfrak{F}} \lambda^*(A) = \mu^*(A)$. If $A$ is a convergence set, then so is $T - A$.

c) Let $A$ be an open or closed convergence set in $T$, and $B$ a convergence set such that $T - B$ is also a convergence set. Then $A \cup B$ and $A \cap B$ are convergence sets, as are their complements.

d) The clan generated by the open convergence sets consists of convergence sets.

e) Let $A$ be a subset of $T$ whose boundary is $\mu$-negligible. Then $A$ is a convergence set.

f) Suppose that $T$ is locally compact or Polish. Show that every compact subset $K$ of $T$ is contained in a compact convergence set (in case $T$ is locally compact, use Exercise 1 b). If $T$ is Polish, use the same exercise to construct a sequence of finite coverings $\mathscr{U}_p$ of $K$ by open subsets of $T$, with $\mu$-negligible boundary and of diameter $< 2^{-p}$; show that $L = \bigcap_p \bigcup_{U \in \mathscr{U}_p} \overline{U}$ is compact and conclude that it is a convergence set by b)).

g) Extend the result of f) to the case of a convergent sequence of measures on a complete metric space.