Mathematische Annalen

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

The Birkhoff integral and the property of Bourgain

  • B. Cascales
  • J. Rodríguez


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.


Banach Space Unit Ball Probability Space Integrable Function Complete Probability 
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© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  1. 1.Departamento de MatemáticasUniversidad de MurciaEspinardoSpain

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