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 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 F. For separation, it is sufficient if 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 F induces the topology of the underlying space, and every bounded set A admits a function in F with values bounded away from zero on A.