Dual representation of convex sets of probability measures on totally bounded spaces

Convex sets of probability measures, frequently encountered in probability theory and statistics, can be transparently analyzed by means of dual representations in a function space. This paper introduces totally bounded spaces, whose structure is defined by a set of bounded real-valued functions, as a general framework for studying such representations. The reinterpretation of classical theorems in this framework clarifies the role of compactness and leads to simple existence criteria. Applications include results on the existence of probability measures satisfying given sets of conditions and an equivalence of consistent preferences and families of probability measures. Moreover, countable additivity of probabilities is seen to be a consequence of elementary consistency assumptions.