Open Access
2016 Boundedly finite measures: separation and convergence by an algebra of functions
Wolfgang Löhr, Thomas Rippl
Electron. Commun. Probab. 21: 1-16 (2016). DOI: 10.1214/16-ECP17


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 $A$.


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Wolfgang Löhr. Thomas Rippl. "Boundedly finite measures: separation and convergence by an algebra of functions." Electron. Commun. Probab. 21 1 - 16, 2016.


Received: 17 March 2016; Accepted: 18 August 2016; Published: 2016
First available in Project Euclid: 9 September 2016

zbMATH: 1348.60140
MathSciNet: MR3548772
Digital Object Identifier: 10.1214/16-ECP17

Primary: 60K35

Keywords: boundedly finite measure , convergence determining , Le Cam theorem , separating , Stone-Weierstrass , vague convergence , weak$^{\#}$-convergence of measures

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