Kantorovich type topologies on spaces of measures and convergence of barycenters

TL;DR

This paper introduces new topologies on measure spaces inspired by Kantorovich norms, analyzes their properties, and establishes convergence criteria for measures and their barycenters, with applications to specific classes like logarithmically concave and stable measures.

Contribution

It defines and studies Kantorovich-type topologies on measure spaces, providing new convergence results for measures and barycenters, especially for special measure classes.

Findings

Kantorovich--Rubinshtein topology coincides with weak topology on certain measure sets.

Provides a sufficient condition for compactness in the Kantorovich topology.

Shows convergence of barycenters for weakly converging logarithmically concave and stable measures.

Abstract

We study two topologies $\tau_{KR}$ and $\tau_K$ on the space of measures on a completely regular space generated by Kantorovich--Rubinshtein and Kantorovich seminorms analogous to their classical norms in the case of a metric space. The Kantorovich--Rubinshtein topology $\tau_{KR}$ coincides with the weak topology on nonnegative measures and on bounded uniformly tight sets of measures. A~sufficient condition is given for the compactness in the Kantorovich topology. We show that for logarithmically concave measures and stable measures weak convergence implies convergence in the Kantorovich topology. We also obtain an efficiently verified condition for convergence of the barycenters of Radon measures from a sequence or net weakly converging on a locally convex space. As an application it is shown that for weakly convergent logarithmically concave measures and stable measures convergence of their barycenters holds without additional conditions. The same is true for measures given by polynomial densities of a fixed degree with respect to logarithmically concave measures.