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

Abstract

We formulate bang-bang, purification, and minimization principles in dual Banach spaces with Gelfand integrals and provide a complete characterization of the saturation property of finite measure spaces. We also present an application of the relaxation technique to large economies with infinite-dimensional commodity spaces, where the space of agents is modeled as a finite measure space. We propose a "relaxation" of large economies, which is regarded as a reasonable convexification of original economies. Under the saturation hypothesis, the relaxation and purification techniques enable us to prove the existence of Pareto optimal allocations without convexity assumptions.

Relaxed large economies with infinite-dimensional commodity spaces: The existence of Walrasian equilibria

“…An germinal notion of saturation already appeared in Kakutani (1944); Maharam (1942). The significance of the saturation property lies in the fact that it is necessary and sufficient for the weak compactness and the convexity of the Bochner integral of a multifunction as well as the Lyapunov convexity theorem in Banach spaces; see Khan and Sagara (2013, 2015, 2016; Podczeck (2008); Sun and Yannelis (2008).…”

“…In the sequel, we may assume without loss of generality that the preference map t → (t) is represented by a Carathéodory function u that is unique up to strictly increasing, continuous transformations. Following Sagara (2016), we introduce the notion of "relaxation" of preferences for large economies. Given a continuous preference (t) on X, its continuous affine extension R (t) to Π(X) is obtained by convexifying (randomizing) the individual utility function u(t, ·) in such a way…”

“…The point is that the existence of the latter is easily established on account of the fact that the relaxed economy is already a convexification of the original economy. This is done through a "purification principle", as in Khan and Sagara (2014); Sagara (2016), a powerful tool whose utility in non-cooperative game theory and statistical decision theory is well-understood. As to the difficulty of the joint continuity of the valuation functional for price-commodity pairs in infinite dimensions when one attempts to apply the fixed point theorem, 6 we adapt the technique employed in Lee (2013); Podczeck (1997) to relaxed demand sets.…”

Fuzzy Core Equivalence in Large Economies: A Role for the Infinite-Dimensional Lyapunov Theorem

“…weak compactness and convexity of the Bochner integral of a multifunction (see Podczeck (2008); Sun and Yannelis (2008)), the bang-bang principle (see Sagara (2014, 2016)), and Fatou's lemma (see ; Khan, Sagara and Suzuki (2016)). For a further generalization of Theorem 2.1 to nonseparable locally convex spaces, see Greinecker and Podczeck (2013); Sagara (2015, 2016); Sagara (2017); Urbinati (2019). Another intriguing characterization of saturation in terms of the existence of Nash equilibria in large games is found in Keisler and Sun (2009).…”

A bang-bang principle for the conditional expectation vector measure

“…However, an infinite dimensional version of such result established by Knowles [29] can be used to establish a purification principle as in Askoura [2]. Furthermore, many works succeeded to recover this property and to establish adequate bang-bang and purification results by using Maharam types and saturated measure spaces; see Greinecker and Podczeck [10], Khan and Sagara [22,23,24,25], Sagara [32] for infinite dimensional Lyapunov convexity theorems, purification processes and applications to the integration of set-valued mappings, equilibrium theory and control systems. Refer to Keisler and Sun [19] and the literature therein for further prevalent studies on this direction.…”