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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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  • 1. G. Birkhoff, Integration of functions with values in a Banach space, Trans. Amer. Math. Soc. 38 (1935), no. 2, 357-378. MR 1501815
  • 2. B. Cascales and J. Rodríguez, The Birkhoff integral and the property of Bourgain, To appear in Math. Ann.
  • 3. D. L. Cohn, Measure theory, Birkhäuser Boston Inc., Boston, MA, 1993, Reprint of the 1980 original. MR 1454121 (98b:28001)
  • 4. L. Di Piazza and D. Preiss, When do McShane and Pettis integrals coincide?, Illinois J. Math. 47 (2003), 1177-1187. MR 2036997
  • 5. J. Diestel and J. J. Uhl, Jr., Vector measures, American Mathematical Society, Providence, R.I., 1977, With a foreword by B. J. Pettis, Mathematical Surveys, No. 15. MR 0453964 (56:12216)
  • 6. M. Fabian, Gâteaux differentiability of convex functions and topology, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons Inc., New York, 1997, Weak Asplund spaces, A Wiley-Interscience Publication. MR 1461271 (98h:46009)
  • 7. M. Fabian, P. Habala, P. Hájek, V. Montesinos Santalucia, J. Pelant, and V. Zizler, Functional analysis and infinite-dimensional geometry, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, vol. 8, Springer-Verlag, New York, 2001. MR 1831176 (2002f:46001)
  • 8. D. H. Fremlin, The generalized McShane integral, Illinois J. Math. 39 (1995), no. 1, 39-67. MR 1299648 (95j:28008)
  • 9. -, The McShane and Birkhoff integrals of vector-valued functions, University of Essex Mathematics Department Research Report 92-10, 1999.
  • 10. D. H. Fremlin and J. Mendoza, On the integration of vector-valued functions, Illinois J. Math. 38 (1994), no. 1, 127-147. MR 1245838 (94k:46083)
  • 11. R. A. Gordon, Riemann integration in Banach spaces, Rocky Mountain J. Math. 21 (1991), no. 3, 923-949. MR 1138145 (92k:28017)
  • 12. V. M. Kadets and L. M. Tseytlin, On ``integration'' of non-integrable vector-valued functions, Mat. Fiz. Anal. Geom. 7 (2000), no. 1, 49-65. MR 1760946 (2001e:28017)
  • 13. K. Musia\l, Topics in the theory of Pettis integration, Rend. Istit. Mat. Univ. Trieste 23 (1991), no. 1, 177-262 (1993), School on Measure Theory and Real Analysis (Grado, 1991). MR 1248654 (94k:46084)
  • 14. -, Pettis integral, Handbook of measure theory, Vols. I, II, North-Holland, Amsterdam, 2002, pp. 531-586. MR 1954622 (2004d:28026)
  • 15. B. J. Pettis, On integration in vector spaces, Trans. Amer. Math. Soc. 44 (1938), no. 2, 277-304. MR 1501970
  • 16. R. S. Phillips, Integration in a convex linear topological space, Trans. Amer. Math. Soc. 47 (1940), 114-145. MR 0002707 (2:103c)
  • 17. A. N. Plichko, On projective resolutions of the identity operator and Markushevich bases, Soviet Math. Dokl. 25 (1982), no. 2, 386-389.
  • 18. M. Talagrand, Pettis integral and measure theory, Mem. Amer. Math. Soc. 51 (1984), no. 307, ix+224. MR 0756174 (86j:46042)
  • 19. M. Valdivia, Simultaneous resolutions of the identity operator in normed spaces, Collect. Math. 42 (1991), no. 3, 265-284 (1992). MR 1203185 (94e:46047)

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Additional Information

José Rodríguez
Affiliation: Departamento de Matemáticas, Universidad de Murcia, 30.100 Espinardo, Murcia, Spain

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.

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