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