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

Kantorovich type topologies on spaces of measures and convergence of barycenters