Let $X$ be a separable metric space, $\mu_{n}$ a sequence of Borel probability measures and $\mathcal{C}$ be a family of sets that is closed under finite unions and interections, and that contains all the balls. If $\mu_{n}(A)$ converges for every $A\in\mathcal{C}$, does there exists a Borel measure $\mu_{\infty}$ such that $\mu_{\infty}(E)=\lim\mu_{n}(E)$ for every $E\in\mathcal{C}?$

From Theorem 4.3 in this paper we can get this result when $X$ is locally compact. Here Sion makes an outer measure and then shows open sets are measurable. His proof definitively uses local compactness.

Does anybody know if the result is true when $X$ is not necessarily locally compact?