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Mathematische Annalen

, Volume 331, Issue 2, pp 259–279 | Cite as

The Birkhoff integral and the property of Bourgain

  • B. Cascales
  • J. Rodríguez
Article

Abstract.

In this paper we study the Birkhoff integral of functions f:Ω→X defined on a complete probability space (Ω,Σ,μ) with values in a Banach space X. We prove that if f is bounded then its Birkhoff integrability is equivalent to the fact that the set of compositions of f with elements of the dual unit ball Z f ={〈x*,f〉:x* ∈ B X* } has the Bourgain property. A non necessarily bounded function f is shown to be Birkhoff integrable if, and only if, Z f is uniformly integrable and has the Bourgain property. As a consequence it turns out that the range of the indefinite integral of a Birkhoff integrable function is relatively norm compact. We characterize the weak Radon-Nikodým property in dual Banach spaces via Birkhoff integrable Radon-Nikodým derivatives. We also point out that a recently introduced notion of unconditional Riemann-Lebesgue integrability coincides with the notion of Birkhoff integrability. Some other applications are given.

Keywords

Banach Space Unit Ball Probability Space Integrable Function Complete Probability 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  1. 1.Departamento de MatemáticasUniversidad de MurciaEspinardoSpain

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