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Boundedly finite measures: separation and convergence by an algebra of functions. (English) Zbl 1348.60140

Summary: 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 \(A\).

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
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