A Glivenko-Cantelli Theorem for almost additive functions

We develop a Glivenko–Cantelli theory for monotone, almost additive functions of i.,i.,d.\ sequences of random variables indexed by~${\mathbb{Z}}^d$. 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.