Fatou's Lemma, Galerkin Approximations and the Existence of Walrasian Equilibria in Infinite Dimensions

This essay has three objectives: (i) to report recent generalizations of Fatou's lemma to multi-functions taking values in a Banach space, and framed in terms of both Bochner and Gelfand integration; (ii) to delineate the importance of Galerkin approximations in Walrasian general equilibrium theory with a continuum of agents and commodities; and thereby (iii) to present two new results on the existence of a Walrasian equilibrium in economies where the continuum of agents is formalized as a saturated measure space.