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We consider random fields indexed by finite subsets of an amenable discretegroup, taking values in the Banach-space of bounded right-continuous functions.The field is assumed to be equivariant, local, coordinate-wise monotone, andalmost additive, with finite range dependence. Using the theory ofquasi-tilings we prove an uniform ergodic theorem, more precisely, thataverages along a Foelner sequence converge uniformly to a limiting function.Moreover we give explicit error estimates for the approximation in the supnorm.

Glivenko-Cantelli Theory, Ornstein-Weiss quasi-tilings, and uniform Ergodic Theorems for distribution-valued fields over amenable groups