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arXiv · 1009.4434

The universal Glivenko-Cantelli property

Ramon van Handel
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Abstract

Let F be a separable uniformly bounded family of measurable functions on a standard measurable space, and let N_{[]}(F,\epsilon,\mu) be the smallest number of \epsilon-brackets in L^1(\mu) needed to cover F. The following are equivalent: 1. F is a universal Glivenko-Cantelli class. 2. N_{[]}(F,\epsilon,\mu)<\infty for every \epsilon>0 and every probability measure \mu. 3. F is totally bounded in L^1(\mu) for every probability measure \mu. 4. F does not contain a Boolean \sigma-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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Ramon van Handel. 2010-09-22. The universal Glivenko-Cantelli property. https://doi.org/10.1007/s00440-012-0416-5

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