On the setwise convergence of sequences of measures

Jean B Lasserre

doi:10.1155/s1048953397000166

Abstract

We consider a sequence {μn} of (nonnegative) measures on a general measurable space (X,ℬ). We establish sufficient conditions for setwise convergence and convergence in total variation.

Highlights

Consider a sequence {#n} of measures on a measurable space (X,%)where X is some topological space
Some important properties can be derived and sufficient conditions ensuring this type of convergence are of interest
The present paper provides two simple sufficient conditions

Summary

Introduction

As opposed to weak* or weak convergence, is a highly desirable and strong property. Some important properties can be derived (for instance, the Vitali-Hahn-Saks Theorem) and sufficient conditions ensuring this type of convergence are of interest. As noted in [2], in contrast to weak or weak* convergence (for instance, in metric spaces), it is in general difficult to exhibit such a property, except if e.g., #n is an increasing or decreasing sequence The present paper provides two simple sufficient conditions. For instance, in a locally compact Hausdorff space, an order-bounded sequence of probability measures that is vaguely or weakly convergent is setwise convergent. We establish a sufficient condition for the convergence in total variation norm that is even a stronger property

Notations and Definitions

Preliminaries

Setwise Convergence