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We formulate bang-bang, purification, and minimization principles in dualBanach spaces with Gelfand integrals and provide a complete characterization ofthe saturation property of finite measure spaces. We also present a newapplication of the relaxation technique to large economies withinfinite-dimensional commodity spaces, where the space of agents is modeled asa finite measure space. We propose a "relaxation" of large economies, which isregarded as a reasonable convexification of original economies. Under thesaturation hypothesis, the relaxation and purification techniques enable us toprove the existence of Pareto optimal allocations without convexityassumptions.

Relaxation and Purification for Nonconvex Variational Problems in Dual Banach Spaces: The Minimization Principle in Saturated Measure Spaces