On a Schur-like property for spaces of measures

TL;DR

This paper establishes a Schur-like property for measures on Polish spaces, showing that certain convergence conditions imply norm convergence, with implications for Markov operators and measure topology equivalences.

Contribution

It introduces a novel Schur-like property for measures, linking weak convergence of integrals to norm convergence in the dual bounded Lipschitz norm.

Findings

Weak convergence of measures implies norm convergence under boundedness conditions.

Equivalence of concepts of equicontinuity in Markov operator theory.

Weak sequential completeness of the space of signed Borel measures.

Abstract

A Banach space has the Schur property when every weakly convergent sequence converges in norm. We prove a Schur-like property for measures: if a sequence of finite signed Borel measures on a Polish space is such that it is bounded in total variation norm and such that for each bounded Lipschitz function the sequence of integrals of this function with respect to these measures converges, then the sequence converges in dual bounded Lipschitz norm or Fortet-Mourier norm to a measure. Moreover, we prove three consequences of this result: the first is equivalence of concepts of equicontinuity in the theory of Markov operators, the second is the derivation of weak sequential completeness of the space of signed Borel measures on Polish spaces from our main result and the third concerns conditions for the coincidence of weak and norm topologies on sets of measures that are bounded in total variation norm with additional properties.