Liapounoff’s vector measure theorem in Banach spaces and applications to general equilibrium theory

Abstract

We present a result on convexity and weak compactness of the range of a vector measure with values in a Banach space, based on the Maharam classification of measure spaces. Our result extends a recent result of Khan and Sagara (Illinois J. Math. 2013). We apply our result to integration of Banach space valued correspondences and to the core-Walras equivalence problem in coalitional exchange economies with an infinite-dimensional commodity space.

Maharam-types and Lyapunov’s theorem for vector measures on Banach spaces

“…An immediate consequence of Proposition 4.1 and Lemma 4.3 is the following version of the Lyapunov theorem, which is a further generalization of [6,10,12]. Theorem 4.3.…”

A convexity result for the range of vector measures with applications to large economies

“…In [20], the authors consider a σ-algebra of sets A and for every infinite cardinal number κ they define a class of κ-atomless measures. The latter consists of σ-additive measure λ : A → [0, 1] such that an equivalent to Lemma 3.3 holds (i.e.…”

“…This idea was then sharpened by Greinecker and Podczeck in [20] and applied to economic models of exchange economies. As observed in [27], this convexity result still holds under the milder assumption that E is a locally convex space, provided that the measure µ admits a real valued control measure, a condition that is always satisfied by Banachspace valued measures.…”

Coalitional extreme desirability in finitely additive exchange economies

“…Moreover, conditions (C-6) and (C-8) can be used to prove the initial part of Lemma 3.2 as well as Lemma 3.3. (c) Zame's framework has been recently reconsidered by Greinecker and Podczeck (2013); as in our work, their aim is to significantly extend the class of Banach lattices on which a coalitional core-Walras equivalence result holds. However, the point of view of the two approaches is substantially different.…”

“…We emphasize that, in the literature, there are other countably additive coalitional models that make use of properness-like conditions in order to obtain core-Walras equivalence theorems. We recall the works of Zame (1986), and the recent one of Greinecker and Podczeck (2013): the last one includes the case of all Banach lattices, at the cost of strengthening some measure theoretic hypotheses. The fact that properness-like assumptions are crucial in order to account for spaces whose positive cone has empty interior is also well emphasized by the recent work of Bhowmik and Graziano (2015), who made use precisely of the above-mentioned conditions for individuals to extend the classical Theorem of Vind (1972) to the case of an ordered Banach space whose positive cone may have empty interior with the presence of atoms in the agents space.…”