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

Abstract

We formulate the saturation property for vector measures in lcHs as a nonseparability condition on the derived Boolean σ-algebras by drawing on the topological structure of vector measure algebras. We exploit a Pettis-like notion of vector integration in lcHs, the Bourbaki-Kluvánek-Lewis integral, to derive an exact version of the Lyapunov convexity theorem in lcHs without the BDS property. We apply our Lyapunov convexity theorem to the bang-bang principle in Lyapunov control systems in lcHs to provide a further characterization of the saturation property.

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

“…The following result is an immediate consequence of Khan and Sagara (2014, Theorem 5.1), whose proof hinges on the Lyapunov convexity theorem in separable Banach spaces obtained in Khan and Sagara (2013) under the saturation hypothesis.…”

“…However, recent work has established the validity of the Lyapunov convexity theorem, and as its corollary, the convexity property of the integral of a multifunction, under the reasonable assumption that the underlying measure space of agents is a saturated space. Both separable Banach spaces and the dual of separable Banach spaces have been considered, and indeed, this saturation property has been shown to be both necessary and sufficient for the results; see Khan and Sagara (2013, 2015, 2016; Podczeck (2008); Sun and Yannelis (2008). It is thus natural then that one looks for a generalization of Aumann's theorem under the assumption of a saturated measure space of agents by exploiting the (exact) convexity of the aggregate demand set in the setting of an infinite-dimensional commodity space even when the individual demand sets are not convex.…”

A Probabilistic Representation of Exact Games on σ-Algebras

“…The saturation property and the Lyapunov convexity theorem in separable Banach spaces are a major apparatus in this paper. Proposition 2.1 (Khan and Sagara [24]). Let E be a separable Banach space.…”

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

“…Theorems 3.1 and 3.2 are another characterization of saturation in terms of (BBP). See [25] for a characterization of saturation in terms of the bang-bang principle with Bochner integrals in separable Banach spaces and [27] for that in terms of the bang-bang principle with Bourbaki-Kluvánek-Lewis integrals in separable locally convex spaces.…”