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

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 $A$.