The Riesz representation in submetric spaces and its probabilistic aspects

Abstract

A submetric space is a topological space equipped with a continuous metric that generates a metric topology (usually) weaker than the original one (e.g., a~separable Hilbert space with the weak topology). We show that a positive linear functional on the space of bounded continuous functions on a submetric space is given as an integral with respect to a tight measure if, and only if, it is continuous in the strict topology. Neither local compactness nor complete regularity is required. We link the equicontinuity of a family of functionals in the strict topology to the uniform tightness of the underlying measures, and we provide a probabilistic representation (Skorokhod's parametrization) for such families. Finally, we identify the finest topology on the space of probability measures on a submetric space in which uniform tightness implies relative compactness. In non-metric spaces this topology is strictly stronger than the topology of weak convergence of probability measures.