A variant of Alaoglu’s theorem for semicontinuous functions

Let X be an arbitrary topological space, and 𝓒( X ) the convex cone of upper semicontinuous bounded functions on X . Further, let 𝓒*( X ) be its dual, i.e., the convex cone of functionals that are additive, positively homogeneous, and monotone. On 𝓒*(X) , we define the weak* topology as the coarsest topology such that, for any f ∈ 𝓒(X) , the evaluation map μ ↦ ∫ f d μ is continuous. Then, the unit ball in 𝓒*(X) is compact in the weak* topology. However, even if X is compact, (i) functionals in 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.