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

  • R. M. Dudley
Part of the Selected Works in Probability and Statistics book series (SWPS)


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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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • R. M. Dudley
    • 1
  1. 1.University of CaliforniaBerkeleyUSA

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