To a mesh function we associate the natural analogue of the Monge-Ampère measure. The latter is shown to be equivalent to the Monge-Ampère 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-Ampère 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-Ampère 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.

对于网格函数，我们将Monge-Ampère度量的自然类似物关联起来。后者被证明等同于凸包络线的蒙格-安培度量。我们证明网格函数的有界凸函数的均匀收敛意味着它们的凸包络的紧子集上的均匀收敛，因此关联的Monge-Ampère测度的弱收敛。我们还给出了使网格函数具有均匀收敛到凸函数的子序列的条件。我们的结果可用于为Monge-Ampère方程的第二个边值问题的一些离散化的收敛提供替代证明，并用于最近提出的后者的离散化。对于均匀有界且在离散水平上满足凸条件的网格函数，我们表明，有一个子序列在紧子集上均匀收敛到凸函数。子序列的网格函数的凸包络也均匀地收敛在紧凑子集上。此外，如果它们与边界上的连续凸函数一致，则表明极限函数强烈满足边界条件。