Electronic Communications in Probability

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

Wolfgang Löhr and Thomas Rippl

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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 $A$.

Article information

Electron. Commun. Probab., Volume 21 (2016), paper no. 60, 16 pp.

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

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Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

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

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Löhr, Wolfgang; Rippl, Thomas. Boundedly finite measures: separation and convergence by an algebra of functions. Electron. Commun. Probab. 21 (2016), paper no. 60, 16 pp. doi:10.1214/16-ECP17. https://projecteuclid.org/euclid.ecp/1473424721

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