  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.
