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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On the existence of Pettis integrable functions which are not Birkhoff integrable
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by José Rodríguez PDF
Proc. Amer. Math. Soc. 133 (2005), 1157-1163 Request permission

Abstract:

Let $X$ be a weakly Lindelöf determined Banach space. We prove that if $X$ is non-separable, then there exist a complete probability space $(\Omega ,\Sigma ,\mu )$ and a bounded Pettis integrable function $f:\Omega \longrightarrow X$ that is not Birkhoff integrable; when the density character of $X$ is greater than or equal to the continuum, then $f$ is defined on $[0,1]$ with the Lebesgue measure. Moreover, in the particular case $X=c_{0}(I)$ (the cardinality of $I$ being greater than or equal to the continuum) the function $f$ can be taken as the pointwise limit of a uniformly bounded sequence of Birkhoff integrable functions, showing that the analogue of Lebesgue’s dominated convergence theorem for the Birkhoff integral does not hold in general.
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Additional Information
  • José Rodríguez
  • Affiliation: Departamento de Matemáticas, Universidad de Murcia, 30.100 Espinardo, Murcia, Spain
  • Email: joserr@um.es
  • Received by editor(s): December 2, 2003
  • Published electronically: September 29, 2004
  • Additional Notes: This research was supported by grant BFM2002-01719 of MCYT and FPU grant of MECD (Spain)
  • Communicated by: N. Tomczak-Jaegermann
  • © Copyright 2004 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 133 (2005), 1157-1163
  • MSC (2000): Primary 28B05, 46G10; Secondary 46B26
  • DOI: https://doi.org/10.1090/S0002-9939-04-07665-8
  • MathSciNet review: 2117218