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The Birkhoff integral and the property of Bourgain

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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.

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Authors and Affiliations

  1. Departamento de Matemáticas, Universidad de Murcia, 30100, Espinardo, Murcia, Spain

    B. Cascales & J. Rodríguez

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  1. B. Cascales
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  2. J. Rodríguez
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Correspondence to B. Cascales.

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Mathematics Subject Classification (2000): 28B05, 46B22, 46G10

Partially supported by the research grant BFM2002-01719 of MCyT (Spain). The second author was supported by a FPU grant of MECD (Spain).

Acknowledgement We gratefully thank Gabriel Vera for lively discussions about some material considered in this paper. We also thank the referee for helpful suggestions that have improved the exposition of this work.

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Cascales, B., Rodríguez, J. The Birkhoff integral and the property of Bourgain. Math. Ann. 331, 259–279 (2005). https://doi.org/10.1007/s00208-004-0581-7

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  • DOI: https://doi.org/10.1007/s00208-004-0581-7

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