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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On the existence of Pettis integrable functions which are not Birkhoff integrable


Author: José Rodríguez
Journal: Proc. Amer. Math. Soc. 133 (2005), 1157-1163
MSC (2000): Primary 28B05, 46G10; Secondary 46B26
Published electronically: September 29, 2004
MathSciNet review: 2117218
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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

DOI: http://dx.doi.org/10.1090/S0002-9939-04-07665-8
PII: S 0002-9939(04)07665-8
Keywords: Pettis integral, Birkhoff integral, McShane integral, dominated convergence theorem, Markushevich basis, weakly Lindel\"of determined Banach space
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
Article copyright: © Copyright 2004 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.