Is the set of probability measures on a compact metric space (weak*-)closed?

Solution 1:

Let $A$ be any Borel set in $S$. By Lusin's theorem, we can choose a sequence of continuous functions $f_m$ with $f_m \to 1_A$ $\mu$-almost everywhere. Moreover, if we set $g_m = \max(\min(f_m, 1), 0)$ then $g_m$ is also continuous, $0 \le g_m \le 1$ and $g_m \to 1_A$ $\mu$-almost everywhere as well.

Since $\int g_m\,d\mu = \lim_{n \to \infty} \int g_m\,d\mu_n$, we have $\int g_m\,d\mu \ge 0$. And by dominated convergence, $\lim_{m \to \infty} \int g_m \,d\mu = \int \lim_{m \to \infty} g_m \,d\mu = \int \mathbf 1_A \,d\mu = \mu(A)$. So $\mu(A) \ge 0$, and we conclude that $\mu$ is a positive measure.