A variant of Alaoglu’s theorem for semicontinuous functions

Let $X$ be an arbitrary topological space, and ${\mathscr C}(X)$ the convex cone of upper semicontinuous bounded functions on $X$. Further, let ${\mathscr C}^*(X)$ be its dual, i.e., the convex cone of functionals that are additive, positively homogeneous, and monotone. On ${\mathscr C}^*(X)$, we define the weak* topology as the coarsest topology such that, for any $f\in {\mathscr C}(X)$, the evaluation map $\mu\mapsto \int f d\mu$ is continuous. Then, the unit ball in ${\mathscr C}^*(X)$ is compact in the weak* topology. However, even if $X$ is compact, (i) functionals in ${\mathscr C}^*(X)$ need not be representable as integrals, and (ii) the space of regular Borel probability measures on $X$ may fail to be compact in the weak* topology. In sum, these observations correct a misrepresentation in the literature and show that the standard approach to establishing mixed-strategy equilibrium existence cannot be easily extended to the non-Hausdorff case.