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Pointwise limits of Birkhoff integrable functions


Author: José Rodríguez
Journal: Proc. Amer. Math. Soc. 137 (2009), 235-245
MSC (2000): Primary 28B05, 46G10
DOI: https://doi.org/10.1090/S0002-9939-08-09589-0
Published electronically: August 13, 2008
MathSciNet review: 2439446
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Abstract: We study the Birkhoff integrability of pointwise limits of sequences of Birkhoff integrable Banach space-valued functions, as well as the convergence of the corresponding integrals. Both norm and weak convergence are considered. We discuss the roles that equi-Birkhoff integrability and the Bourgain property play in these problems. Incidentally, a convergence theorem for the Pettis integral with respect to the norm topology is presented.


References [Enhancements On Off] (What's this?)

  • 1. M. Balcerzak and M. Potyrała, Convergence theorems for the Birkhoff integral, to appear in Czechoslovak Math. J.
  • 2. R. G. Bartle, A modern theory of integration, Graduate Studies in Mathematics, vol. 32, American Mathematical Society, Providence, RI, 2001. MR 1817647 (2002d:26001)
  • 3. G. Birkhoff, Integration of functions with values in a Banach space, Trans. Amer. Math. Soc. 38 (1935), no. 2, 357-378. MR 1501815
  • 4. B. Cascales and J. Rodríguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540-560, Special issue dedicated to John Horváth. MR 2088679 (2005f:26021)
  • 5. -, The Birkhoff integral and the property of Bourgain, Math. Ann. 331 (2005), no. 2, 259-279. MR 2115456 (2006i:28006)
  • 6. J. Diestel and J. J. Uhl, Jr., Vector measures, with a foreword by B. J. Pettis, Mathematical Surveys, No. 15, American Mathematical Society, Providence, RI, 1977. MR 0453964 (56:12216)
  • 7. D. H. Fremlin, The McShane and Birkhoff integrals of vector-valued functions, University of Essex Mathematics Department Research Report 92-10, version of 18.05.07 available at http://www.essex.ac.uk/maths/staff/fremlin/preprints.htm.
  • 8. -, The generalized McShane integral, Illinois J. Math. 39 (1995), no. 1, 39-67. MR 1299648 (95j:28008)
  • 9. -, Measure theory. Volume 2: Broad foundations, Torres Fremlin, Colchester, 2001.
  • 10. R. A. Gordon, The integrals of Lebesgue, Denjoy, Perron, and Henstock, Graduate Studies in Mathematics, vol. 4, American Mathematical Society, Providence, RI, 1994. MR 1288751 (95m:26010)
  • 11. J. Kurzweil and S. Schwabik, McShane equi-integrability and Vitali's convergence theorem, Math. Bohem. 129 (2004), no. 2, 141-157. MR 2073511 (2005f:26022)
  • 12. K. Musiał, 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)
  • 13. -, Pettis integral, Handbook of Measure Theory, Vol. I, II, North-Holland, Amsterdam, 2002, pp. 531-586. MR 1954622 (2004d:28026)
  • 14. B. J. Pettis, On integration in vector spaces, Trans. Amer. Math. Soc. 44 (1938), no. 2, 277-304. MR 1501970
  • 15. L. H. Riddle and E. Saab, On functions that are universally Pettis integrable, Illinois J. Math. 29 (1985), no. 3, 509-531. MR 786735 (86i:28012)
  • 16. J. Rodríguez, Convergence theorems for the Birkhoff integral, to appear in Houston J. Math., preprint available at http://personales.upv.es/jorodrui.
  • 17. -, On the existence of Pettis integrable functions which are not Birkhoff integrable, Proc. Amer. Math. Soc. 133 (2005), no. 4, 1157-1163. MR 2117218 (2005k:28021)
  • 18. -, Universal Birkhoff integrability in dual Banach spaces, Quaest. Math. 28 (2005), no. 4, 525-536. MR 2182459 (2006g:28029)
  • 19. -, On integration of vector functions with respect to vector measures, Czechoslovak Math. J. 56(131) (2006), no. 3, 805-825. MR 2261655 (2007j:28019)
  • 20. -, The Bourgain property and convex hulls, Math. Nachr. 280 (2007), no. 11, 1302-1309. MR 2337347
  • 21. -, Spaces of vector functions that are integrable with respect to vector measures, J. Aust. Math. Soc. 82 (2007), no. 1, 85-109. MR 2301972
  • 22. A. P. Solodov, On the limits of the generalization of the Kolmogorov integral, Mat. Zametki 77 (2005), no. 2, 258-272. MR 2157094 (2006c:46037)
  • 23. M. Talagrand, Pettis integral and measure theory, Mem. Amer. Math. Soc. 51 (1984), no. 307. MR 756174 (86j:46042)

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Additional Information

José Rodríguez
Affiliation: Instituto Universitario de Matemática Pura y Aplicada, Universidad Politécnica de Valencia, Camino de Vera s/n, 46022 Valencia, Spain
Email: jorodrui@mat.upv.es

DOI: https://doi.org/10.1090/S0002-9939-08-09589-0
Keywords: Birkhoff integral, Pettis integral, Bourgain property, pointwise limit, convergence theorem
Received by editor(s): January 8, 2008
Published electronically: August 13, 2008
Additional Notes: This research was supported by the Spanish grant MTM2005-08379 (MEC and FEDER)
Communicated by: Tatiana Toro
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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