  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.
