  We show that if $\mathcal{X}$ is a complete separable metric space and
$\mathcal{C}$ is a countable family of Borel subsets of $\mathcal{X}$ with
finite VC dimension, then, for every stationary ergodic process with values in
$\mathcal{X}$, the relative frequencies of sets $C\in\mathcal{C}$ converge
uniformly to their limiting probabilities. Beyond ergodicity, no assumptions
are imposed on the sampling process, and no regularity conditions are imposed
on the elements of $\mathcal{C}$. The result extends existing work of Vapnik
and Chervonenkis, among others, who have studied uniform convergence for i.i.d.
and strongly mixing processes. Our method of proof is new and direct: it does
not rely on symmetrization techniques, probability inequalities or mixing
conditions. The uniform convergence of relative frequencies for VC-major and
VC-graph classes of functions under ergodic sampling is established as a
corollary of the basic result for sets.
