The universal Glivenko-Cantelli property

Let F be a separable uniformly bounded family of measurable functions on a standard measurable space (X, X, 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: F is a universal Glivenko-Cantelli class. N-[](F, epsilon, mu) < infinity for every epsilon > 0 and every probability measure mu. F is totally bounded in L-1(mu) for every probability measure mu. 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.