Boundedly finite measures: Separation and convergence by an algebra of functions

TL;DR

This paper establishes conditions under which an algebra of functions can distinguish and determine the convergence of boundedly finite measures on separable metric and Souslin spaces, advancing measure theory and functional analysis.

Contribution

It provides new criteria for separation and weak$^ ext{#}$-convergence determination of measures using algebras of functions, extending existing theoretical frameworks.

Findings

Conditions for an algebra to separate measures include point separation and vanishing nowhere.

For convergence determination, the algebra must induce the space topology and have functions bounded away from zero on bounded sets.

Results apply to measures on separable metric and Souslin spaces, broadening applicability.

Abstract

We prove general results about separation and weak$^\#$-convergence of boundedly finite measures on separable metric spaces and Souslin spaces. More precisely, we consider an algebra of bounded real-valued, or more generally a $*$-algebra $\mathcal{F}$ of bounded complex-valued functions and give conditions for it to be separating or weak$^\#$-convergence determining for those boundedly finite measures that integrate all functions in $\mathcal{F}$. For separation, it is sufficient if $\mathcal{F}$ separates points, vanishes nowhere, and either consists of only countably many measurable functions, or of arbitrarily many continuous functions. For convergence determining, it is sufficient if $\mathcal{F}$ induces the topology of the underlying space, and every bounded set $A$ admits a function in $\mathcal{F}$ with values bounded away from zero on $A$.