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

TL;DR

This paper explores advanced mathematical tools like Fatou's lemma and Galerkin approximations to establish the existence of Walrasian equilibria in complex infinite-dimensional economic models.

Contribution

It generalizes Fatou's lemma for Banach space-valued functions and applies Galerkin methods to prove equilibrium existence in economies with infinite agents and commodities.

Findings

Generalized Fatou's lemma for Banach space-valued functions.

Established the importance of Galerkin approximations in infinite-dimensional equilibrium models.

Proved new existence results for Walrasian equilibria in saturated measure spaces.

Abstract

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.