The universal Glivenko-Cantelli property
arXiv:1009.4434 · doi:10.1007/s00440-012-0416-5
Abstract
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.
26 pages