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Integration in Hilbert generated Banach spaces

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Abstract

We prove that McShane and Pettis integrability are equivalent for functions taking values in a subspace of a Hilbert generated Banach space. This generalizes simultaneously all previous results on such equivalence. On the other hand, for any super-reflexive generated Banach space having density character greater than or equal to the continuum, we show that Birkhoff integrability lies strictly between Bochner and McShane integrability. Finally, we give a ZFC example of a scalarly null Banach space-valued function (defined on a Radon probability space) which is not McShane integrable.

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Author notes

  1. José Rodríguez

    Present address: Departamento de Matemática Aplicada, Facultad de Informática, Universidad de Murcia, 30100, Espinardo (Murcia), Spain

Authors and Affiliations

  1. Laboratoire Bordelais d’Analyse et Geométrie, Institut de Mathématiques de Bordeaux, Université de Bordeaux 1, 351 cours de la Libération, Talence Cedex, 33405, France

    Robert Deville

  2. Instituto Universitario de Matemática Pura y Aplicada, Universidad Politécnica de Valencia, Camino de Vera s/n, 46022, Valencia, Spain

    José Rodríguez

Authors
  1. Robert Deville
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  2. José Rodríguez
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Correspondence to Robert Deville.

Additional information

The second-named author was supported by MEC and FEDER (project MTM2005-08379).

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Deville, R., Rodríguez, J. Integration in Hilbert generated Banach spaces. Isr. J. Math. 177, 285–306 (2010). https://doi.org/10.1007/s11856-010-0047-4

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