  For any family of measurable sets in a probability space, we show that either
(i) the family has infinite Vapnik-Chervonenkis (VC) dimension or (ii) for
every epsilon > 0 there is a finite partition pi such the pi-boundary of each
set has measure at most epsilon. Immediate corollaries include the fact that a
family with finite VC dimension has finite bracketing numbers, and satisfies
uniform laws of large numbers for every ergodic process. From these
corollaries, we derive analogous results for VC major and VC graph families of
functions.
