Asymptotic distribution and weak convergence on compact Riemannian manifolds
type: Item
instance of: scholarly article
title: Asymptotic distribution and weak convergence on compact Riemannian manifolds
zbMATH Open document ID: 0717.28001
DOI: 10.1007/BF01301674
author: Martin Blümlinger
published in: Monatshefte für Mathematik
publication date: 1990-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/178494
review text: 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.
Mathematics Subject Classification ID: 28A33
Mathematics Subject Classification ID: 60B10
Mathematics Subject Classification ID: 11K38
zbMATH DE Number: 4181678
zbMATH Keywords: weak convergence
zbMATH Keywords: asymptotic distribution
zbMATH Keywords: compact Riemannian manifold
zbMATH Keywords: discrepancy of two measures
zbMATH Keywords: absolutely continuous measure
zbMATH Keywords: Kantorovich metric
Wikidata QID: Q115393725
MaRDI profile type: Publication
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OpenAlex ID: W2382294934
description: scientific article; zbMATH DE number 4181678