Choquet type L-1-spaces of a vector capacity

Abstract

[EN] Given a set function Lambda with values in a Banach space X, we construct an integration theory for scalar functions with respect to Lambda by using duality on Xand Choquet scalar integrals. Our construction extends the classical Bartle-Dunford-Schwartz integration for vector measures. Since just the minimal necessary conditions on Lambda are required, several L-1-spaces of integrable functions associated to Lambda appear in such a way that the integration map can be defined in them. We study the properties of these spaces and how they are related. We show that the behavior of the L-1-spaces and the integration map can be improved in the case when Xis an order continuous Banach lattice, providing new tools for (non-linear) operator theory and information sciences. (C) 2017 Elsevier B.V. All rights reserved.