On the structure of bounded smooth measures associated with a quasi-regular Dirichlet form

TL;DR

This paper characterizes bounded measures related to quasi-regular Dirichlet forms, showing they can be decomposed into an integrable function and a bounded linear functional, aiding in understanding measures that do not charge sets of zero capacity.

Contribution

It provides a new decomposition theorem for bounded measures associated with quasi-regular Dirichlet forms, clarifying their structure and properties.

Findings

Measures charging no zero-capacity sets can be decomposed explicitly.

The decomposition involves an integrable function and a bounded linear functional.

Examples illustrate the applicability of the decomposition to various Dirichlet forms.

Abstract

We consider a quasi-regular Dirichlet form. We show that a bounded signed measure charges no set of zero capacity associated with the form if and only if the measure can be decomposed into the sum of an integrable function and a bounded linear functional on the domain of the form. The decomposition allows one to describe explicitly the set of bounded measures charging no sets of zero capacity for interesting classes of Dirichlet forms. By way of illustration, some examples are given.