Uniform measures and countably additive measures

TL;DR

This paper explores the properties of uniform measures, their relationship with countably additive measures, and characterizes functionals continuous on certain function spaces within the context of uniform spaces.

Contribution

It establishes conditions under which countably additive measures are uniform measures and characterizes sequentially continuous functionals as uniform measures on separable modifications.

Findings

Every countably additive measure is a uniform measure if all cardinals have measure zero.

Sequentially continuous functionals on bounded uniformly equicontinuous sets are exactly uniform measures on the separable modification.

Provides a characterization of uniform measures in the context of uniform spaces.

Abstract

Uniform measures are defined as the functionals on the space of bounded uniformly continuous functions that are continuous on bounded uniformly equicontinuous sets. If every cardinal has measure zero then every countably additive measure is a uniform measure. The functionals sequentially continuous on bounded uniformly equicontinuous sets are exactly uniform measures on the separable modification of the underlying uniform space.