Asymptotic distribution and weak convergence on compact Riemannian manifolds (Q753980)

On a compact Riemannian manifold the discrepancy of two measures \(\mu\), \(\nu\) is defined as sup\(| \mu (B)-\nu (B)|\), where the supremum is taken over all geodesic balls B in the manifold. It is shown that a sequence of bounded measures converges weakly to an absolutely continuous measure (with respect to the volume measure of M) if and only if the discrepancy converges to 0, and that the geodesic balls with vanishing \(\nu\)-measure of the boundary constitute a convergence determining class of sets for a positive measure \(\nu\). Estimates for the distance of a probability measure to the normalized volume measure on M with respect to the Kantorovich metric in terms of the discrepancy are obtained.