A Glivenko-Cantelli theorem for almost additive functions on lattices

We develop a Glivenko-Cantelli theory for monotone, almost additive functions of i.i.d. sequences of random variables indexed by  Zd. Under certain conditions on the random sequence, short range correlations are allowed as well. We have an explicit error estimate, consisting of a probabilistic and a geometric part. We apply the results to yield uniform convergence for several quantities arising naturally in statistical physics.