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Spaces of vector functions that are integrable with respect to vector measures

Part of: Linear function spaces and their duals Topological linear spaces and related structures Set functions, measures and integrals with values in abstract spaces Measures, integration, derivative, holomorphy

Published online by Cambridge University Press:  09 April 2009

José Rodríguez
José Rodríguez
Affiliation:
Departamento de Matemáticas Universidad de Murcia30.100 Espinardo MurciaSpain e-mail: joserr@um.es
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Abstract

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We study the normed spaces of (equivalence classes of) Banach space-valued functions that are Dobrakov, S* or McShane integrable with respect to a Banach space-valued measure, where the norm is the natural one given by the total semivariation of the indefinite integral. We show that simple functions are dense in these spaces. As a consequence we characterize when the corresponding indefinite integrals have norm relatively compact range. On the other hand, we also determine when these spaces are ultrabornological. Our results apply to conclude, for instance, that the spaces of Birkhoff (respectively McShane) integrable functions defined on a complete (respectively quasi-Radon) probability space, endowed with the Pettis norm, are ultrabornological.

Keywords

Dobrakov integral; S*-integral; McShane integral; Birkhoff integral; ultrabornological space; barrelled space.

MSC classification

Secondary: 28B05: Vector-valued set functions, measures and integrals 46G10: Vector-valued measures and integration 46A08: Barrelled spaces, bornological spaces 46E40: Spaces of vector- and operator-valued functions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 82 , Issue 1 , February 2007 , pp. 85 - 109
Copyright
Copyright © Australian Mathematical Society 2007

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