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Some measure theoretical characterizations of separability of metric spaces. (English) Zbl 0822.54024
Let \((X,d)\) denote a metric space with metric \(d\) and Borel \(\sigma\)- algebra \({\mathfrak B}_ X\). The main result of this note concerns the following characterization of separability of \((X,d)\):
Theorem. Both of the following two conditions are equivalent to separability of the metric space \((X,d)\): (i) \({\mathfrak B}_ X\) is countably generated and any closed, uncountable subset \(Y\) of \(X\), such that \((Y,d)\) is complete, has cardinality of the continuum. (ii) \({\mathfrak B}_ X\cap Y\) admits for any closed, uncountable subset \(Y\) of \(X\), such that \((Y,d)\) is complete, a continuous probability measure.

MSC:
54E35 Metric spaces, metrizability
54D65 Separability of topological spaces
54E70 Probabilistic metric spaces
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