  To a mesh function we associate the natural analogue of the Monge-Ampere
measure. The latter is shown to be equivalent to the Monge-Ampere measure of
the convex envelope. We prove that the uniform convergence to a bounded convex
function of mesh functions implies the uniform convergence on compact subsets
of their convex envelopes and hence the weak convergence of the associated
Monge-Ampere measures. We also give conditions for mesh functions to have a
subsequence which converges uniformly to a convex function. Our result can be
used to give alternate proofs of the convergence of some discretizations for
the second boundary value problem for the Monge-Ampere equation and was used
for a recently proposed discretization of the latter. For mesh functions which
are uniformly bounded and satisfy a convexity condition at the discrete level,
we show that there is a subsequence which converges uniformly on compact
subsets to a convex function. The convex envelopes of the mesh functions of the
subsequence also converge uniformly on compact subsets. If in addition they
agree with a continuous convex function on the boundary, the limit function is
shown to satisfy the boundary condition strongly.
