How regular can maxitive measures be?

TL;DR

This paper investigates the regularity properties of domain-valued maxitive measures on topological spaces, providing characterizations, conditions for densities, and decompositions based on the space's structure.

Contribution

It offers new characterizations of regular maxitive measures, links regularity to complete maxitivity, and presents a decomposition theorem under certain conditions.

Findings

Regular maxitive measures are characterized based on the topological space.

Every regular maxitive measure is completely maxitive, ensuring the existence of a density.

Outer-continuous maxitive measures can be decomposed into regular and vanishing parts.

Abstract

We examine domain-valued maxitive measures defined on the Borel subsets of a topological space. Several characterizations of regularity of maxitive measures are proved, depending on the structure of the topological space. Since every regular maxitive measure is completely maxitive, this yields sufficient conditions for the existence of a cardinal density. We also show that every outer-continuous maxitive measure can be decomposed as the supremum of a regular maxitive measure and a maxitive measure that vanishes on compact subsets under appropriate conditions.