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The universal Glivenko-Cantelli property
Ramon van Handel · 2010-09-23 · via math.ST updates on arXiv.org

Let F be a separable uniformly bounded family of measurable functions on a standard measurable space, and let N_{[]}(F,ε,μ) be the smallest number of ε-brackets in L^1(μ) needed to cover F. The following are equivalent: 1. F is a universal Glivenko-Cantelli class. 2. N_{[]}(F,ε,μ)<\infty for every ε>0 and every probability measure μ. 3. F is totally bounded in L^1(μ) for every probability measure μ. 4. 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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