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