Please use this identifier to cite or link to this item: https://hdl.handle.net/10419/335170 
Year of Publication: 
2025
Series/Report no.: 
Working Paper No. 483
Publisher: 
University of Zurich, Department of Economics, Zurich
Abstract: 
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 7Ç R f dµ 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 mixedstrategy equilibrium existence cannot be easily extended to the non-Hausdorff case.
Subjects: 
Semicontinuous functions
Positively homogeneous, additive, and monotone functionals
Weak* topology
Alaoglu's Theorem
Compactness in convex cones
Persistent Identifier of the first edition: 
Document Type: 
Working Paper

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