Weak-star compactness of probability measures

If $K=\mathrm{\varnothing }$, no measure can have total mass one, so the probability space is empty and compact. Hence assume $K\mathrm{\ne }\mathrm{\varnothing }$. By [F5], compactness makes $K$ locally compact, and it is Hausdorff by hypothesis. Every continuous function on compact $K$ has compact support and compact closed superlevel sets, so [F6] gives $C_c(K)=C_0(K)=C(K)$.

A regular Borel probability $\mu$ defines $L_{\mu }(g)={\int }_{K}gd\mu$. For real or complex $g$, $\mathrm{\mid }{L}_{\mu }(g)\mathrm{\mid }\le \parallel g{\parallel }_{\mathrm{\infty }}\mu (K)=\parallel g{\parallel }_{\mathrm{\infty }}$, while $L_{\mu }(1)=1$; hence $L_{\mu }\in B_{C(K{)}^{\ast }}$ and $\parallel L_{\mu }\parallel =1$. It is positive on nonnegative real-valued $g$.

Conversely, let $L\in B_{C(K{)}^{\ast }}$ satisfy $L(1)=1$ and $L(g)\in [0,\mathrm{\infty })$ for every nonnegative real-valued $g\in C(K)$. In the real case [F2] represents $L$ by a finite regular measure $\mu$ in the sense of [F4], and $\mu (K)=L(1)=1$. In the complex case restrict $L$ to real-valued functions, apply [F2], and use complex linearity to recover $L(g+ih)=\int gd\mu +i\int hd\mu$; uniqueness in [F3] identifies this same positive probability measure. This is the use of Dependent Choice; the ultrafilter lemma is used separately in [F1].

Thus probabilities correspond exactly to the slice $S$ of $B_{C(K{)}^{\ast }}$ cut out by $L(1)=1$ and by $L(g)\in [0,\mathrm{\infty })$ for all nonnegative real-valued $g$. Each condition is weak-star closed because it is the inverse image of the closed set $\{1\}$ or $[0,\mathrm{\infty })$ under one evaluation; arbitrary intersections remain closed.

By [F1], the dual ball is weak-star compact under the ultrafilter lemma. Hence its closed subset $S$ is compact by [F7]. Under the two representation directions above, the weak-star topology is exactly convergence of $\int gd\mu$ for every continuous test function $g$, so the probability measures are compact in the asserted topology.

The initial reduction proves the empty case, and the closed-slice construction proves both scalar-field cases for nonempty compact Hausdorff $K$.