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Positivity

, Volume 13, Issue 1, pp 61–87 | Cite as

Vector measures: where are their integrals?

  • Guillermo P. CurberaEmail author
  • Olvido Delgado
  • Werner J. Ricker
Article

Abstract

Let ν be a vector measure with values in a Banach space Z. The integration map \(I_\nu: L^1(\nu)\to Z\), given by \(f\mapsto \int f\,d\nu\) for fL1(ν), always has a formal extension to its bidual operator \(I_\nu^{**}: L^1(\nu)^{**}\to Z^{**}\). So, we may consider the “integral” of any element f** of L1(ν)** as I ν ** (f**). Our aim is to identify when these integrals lie in more tractable subspaces Y of Z**. For Z a Banach function space X, we consider this question when Y is any one of the subspaces of X** given by the corresponding identifications of X, X′′ (the Köthe bidual of X) and X* (the topological dual of the Köthe dual of X). Also, we consider certain kernel operators T and study the extended operator I ν ** for the particular vector measure ν defined by ν(A) := T A ).

Mathematics Subject Classification (2000)

Primary 46G10, 47B38 Secondary 46B42, 47G10 

Keywords

Banach lattices and function spaces vector measure integration map Duality 

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Copyright information

© Birkhäuser Verlag Basel/Switzerland 2008

Authors and Affiliations

  • Guillermo P. Curbera
    • 1
    Email author
  • Olvido Delgado
    • 2
  • Werner J. Ricker
    • 3
  1. 1.Facultad de MatemáticasUniversidad de SevillaSevillaSpain
  2. 2.Instituto Universitario de Matemática Pura y AplicadaUniversidad Politécnica de ValenciaValenciaSpain
  3. 3.Math.-Geogr. FakultätKatholische Universität Eichstätt-IngolstadtEichstättGermany

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