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

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**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.## References

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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