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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
##### Keywords:
separability; probability measure
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##### References:
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