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THE FATOU COMPLETION OF A FRÉCHET FUNCTION SPACE AND APPLICATIONS

Part of: Set functions, measures and integrals with values in abstract spaces Connections with other structures, applications Linear function spaces and their duals Topological linear spaces and related structures

Published online by Cambridge University Press:  08 December 2009

R. DEL CAMPO  and
W. J. RICKER
R. DEL CAMPO
Affiliation:
Dpto. Matemática Aplicada I, EUITA, Ctra. de Utrera Km. 1, E–41013 Sevilla, Spain (email: rcampo@us.es)
W. J. RICKER*
Affiliation:
Math.-Geogr. Fakultät, Katholische Universität Eichstätt-Ingolstadt, D–85072 Eichstätt, Germany (email: werner.ricker@ku-eichstaett.de)
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Abstract

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Given a metrizable locally convex-solid Riesz space of measurable functions we provide a procedure to construct a minimal Fréchet (function) lattice containing it, called its Fatou completion. As an application, we obtain that the Fatou completion of the space L1(ν) of integrable functions with respect to a Fréchet-space-valued measure ν is the space L1w(ν) of scalarly ν-integrable functions. Further consequences are also given.

Keywords

Fréchet space (lattice)vector measureFatou propertyLebesgue topologyscalarly integrable function

MSC classification

Secondary: 28B05: Vector-valued set functions, measures and integrals 46A40: Ordered topological linear spaces, vector lattices 46E30: Spaces of measurable functions ($L^p$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 54H12: Topological lattices, etc.
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 88 , Issue 1 , February 2010 , pp. 49 - 60
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2009

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