Weak convergence of measures and a limit theorem for correspondences without closed graphs

A sequence of measurable, closed-valued correspondences between metric spaces converges to a correspondence with the same properties.These correspondences need not have closed graphs.Each element of a convergent sequence of measures gives full measure to the graph of the associated correspondence.Under a uniform absolute continuity condition, the limit measure gives full measure to the limit correspondence.An application to the upper hemicontinuity of the Nash equilibrium correspondence of large games is provided.This discussion is also relevant to the upper hemicontinuity of the Walrasian equilibrium correspondence in general equilibrium theory.