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The universal Glivenko–Cantelli property
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  • Published: 12 February 2012

The universal Glivenko–Cantelli property

  • Ramon van Handel1 

Probability Theory and Related Fields volume 155, pages 911–934 (2013)Cite this article

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Abstract

Let \({\mathcal{F}}\) be a separable uniformly bounded family of measurable functions on a standard measurable space \({(X, \mathcal{X})}\), and let \({N_{[]}(\mathcal{F}, \varepsilon, \mu)}\) be the smallest number of \({\varepsilon}\) -brackets in L 1(μ) needed to cover \({\mathcal{F}}\). The following are equivalent:

  1. 1.

    \({\mathcal{F}}\) is a universal Glivenko–Cantelli class.

  2. 2.

    \({N_{[]}(\mathcal{F},\varepsilon,\mu) < \infty}\) for every \({\varepsilon > 0}\) and every probability measure μ.

  3. 3.

    \({\mathcal{F}}\) is totally bounded in L 1(μ) for every probability measure μ.

  4. 4.

    \({\mathcal{F}}\) does not contain a Boolean σ-independent sequence.

It follows that universal Glivenko–Cantelli classes are uniformity classes for general sequences of almost surely convergent random measures.

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Authors and Affiliations

  1. Princeton University, Sherrerd Hall, Room 227, Princeton, NJ, 08544, USA

    Ramon van Handel

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  1. Ramon van Handel
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Correspondence to Ramon van Handel.

Additional information

This work was partially supported by NSF grant DMS-1005575.

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van Handel, R. The universal Glivenko–Cantelli property. Probab. Theory Relat. Fields 155, 911–934 (2013). https://doi.org/10.1007/s00440-012-0416-5

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  • Received: 15 March 2011

  • Revised: 20 December 2011

  • Published: 12 February 2012

  • Issue Date: April 2013

  • DOI: https://doi.org/10.1007/s00440-012-0416-5

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Keywords

  • Universal Glivenko–Cantelli classes
  • Uniformity classes
  • Uniform convergence of random measures
  • Entropy with bracketing
  • Boolean independence

Mathematics Subject Classification (2000)

  • 60F15
  • 60B10
  • 41A46
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