On a convergence of positive continuous additive functionals in terms of their smooth measures

TL;DR

This paper investigates the convergence of positive continuous additive functionals through their smooth measures, establishing a compactness result for the Revuz map and introducing a new metric on measures of finite energy.

Contribution

It introduces a metric on measures of finite energy and proves a compactness result for the Revuz map related to additive functionals.

Findings

Established compactness of the Revuz map under locally uniform convergence

Introduced a new metric on measures of finite energy

Provided examples demonstrating convergence in terms of smooth measures

Abstract

A compactness of the Revuz map is established in the sense that the locally uniform convergence of a sequence of positive continuous additive functionals is derived in terms of their smooth measures. To this end, we first introduce a metric on the space of measures of finite energy integrals and show some structures of the metric. Then, we show the compactness and give some examples of positive continuous additive functionals that the convergence holds in terms of the associated smooth measures.