  Let F be a family of Borel measurable functions on a complete separable
metric space. The gap (or fat-shattering) dimension of F is a combinatorial
quantity that measures the extent to which functions f in F can separate finite
sets of points at a predefined resolution gamma > 0. We establish a connection
between the gap dimension of F and the uniform convergence of its sample
averages under ergodic sampling. In particular, we show that if the gap
dimension of F at resolution gamma > 0 is finite, then for every ergodic
process the sample averages of functions in F are eventually within 10 gamma of
their limiting expectations uniformly over the class F. If the gap dimension of
F is finite for every resolution gamma > 0 then the sample averages of
functions in F converge uniformly to their limiting expectations. We assume
only that F is uniformly bounded and countable (or countably approximable). No
smoothness conditions are placed on F, and no assumptions beyond ergodicity are
placed on the sampling processes. Our results extend existing work for i.i.d.
processes.
