Topological Stationarity and Precompactness of Probability Measures

Abstract

We prove the precompactness of a collection of Borel probability measures over an arbitrary metric space precisely under a new legitimate notion, which we term \textit{topological stationarity}, regulating the sequential behavior of Borel probability measures directly in terms of the open sets. Thus the important direct part of Prokhorov's theorem, which permeates the weak convergence theory, admits a new version with the original and sole assumption --- tightness --- replaced by topological stationarity. Since, as will be justified, our new condition is not vacuous and is logically independent of tightness, our result deepens the understanding of the connection between precompactness of Borel probability measures and metric topologies.