Selected Works of R.M. Dudley pp 5-22 | Cite as

# Weak Convergence of Probabilities on Nonseparable Metric Spaces and Empirical Measures on Euclidean Spaces

## Abstract

It is known that under certain mild set-theoretic assumptions, a finite, countably additive measure defined on all Borel sets of a metric space is concentrated in a separable subspace (Marczewski and Sikorski [8]). However, there are interesting probability measures on metric spaces not concentrated in separable subspaces. In this paper, we consider countably additive probability measures on the smallest *σ*-field containing the open balls of a metric space. This *σ*-field is the Borel field for a separable space, but is smaller in general. A probability measure on it need not be confined to a separable subspace.

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