On Radon Barycenters of Measures on Spaces of Measures

Abstract

We study metrizability of compact sets in spaces of Radon measures with the weak topology. It is shown that if all compacta in a given completely regular topological space are metrizable, then every uniformly tight compact set in the space of Radon measures on this space is also metrizable. It is proved that the property that compact sets of measures on a given space are metrizable is preserved for products of this space with spaces that can be embedded into separable metric spaces. In addition, we construct a Radon probability measure on the space of Radon probability measures on a completely regular space such that its barycenter is not a Radon measure.