Exercise 4, § 5

Let T be a completely regular space, t a point of T, and $\mathscr{U}$ a set of Borel subsets of T that generates the filter of neighborhoods of t. Let I be a set equipped with a filter $\mathfrak{F}$, and let $(\mu_i)_{i \in I}$ be a family of bounded positive measures of total mass 1 on T. In order that $\lim_{i, \mathfrak{F}} \mu_i = \varepsilon_t$, it is necessary and sufficient that $\lim_{i, \mathfrak{F}} \mu_i(U) = 1$ for every $U \in \mathscr{U}$.