The universal Glivenko-Cantelli property

TL;DR

This paper characterizes universal Glivenko-Cantelli classes through equivalent conditions involving bracketing numbers, total boundedness, and independence, establishing their role as uniformity classes for convergent random measures.

Contribution

It provides a comprehensive equivalence characterization of universal Glivenko-Cantelli classes, connecting measure-theoretic, combinatorial, and topological properties.

Findings

Universal Glivenko-Cantelli classes are characterized by finite bracketing numbers.

Such classes are totally bounded in L^1(or every probability measure.

They do not contain Boolean rom which they serve as uniformity classes.

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.