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

DOI:
https://doi.org/10.1090/S0002-9939-04-07665-8

Published electronically:
September 29, 2004

MathSciNet review:
2117218

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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

Keywords:
Pettis integral,
Birkhoff integral,
McShane integral,
dominated convergence theorem,
Markushevich basis,
weakly Lindelöf 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.