Pointwise limits of Birkhoff integrable functions

Rodríguez, José

doi:10.1090/S0002-9939-08-09589-0

Pointwise limits of Birkhoff integrable functions
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by José Rodríguez

Proc. Amer. Math. Soc. 137 (2009), 235-245

DOI: https://doi.org/10.1090/S0002-9939-08-09589-0

Published electronically: August 13, 2008

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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

M. Balcerzak and M. Potyrała, Convergence theorems for the Birkhoff integral, to appear in Czechoslovak Math. J.
Robert G. Bartle, A modern theory of integration, Graduate Studies in Mathematics, vol. 32, American Mathematical Society, Providence, RI, 2001. MR 1817647, DOI 10.1090/gsm/032
Garrett Birkhoff, Integration of functions with values in a Banach space, Trans. Amer. Math. Soc. 38 (1935), no. 2, 357–378. MR 1501815, DOI 10.1090/S0002-9947-1935-1501815-3
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, DOI 10.1016/j.jmaa.2004.03.026
B. Cascales and J. Rodríguez, The Birkhoff integral and the property of Bourgain, Math. Ann. 331 (2005), no. 2, 259–279. MR 2115456, DOI 10.1007/s00208-004-0581-7
J. Diestel and J. J. Uhl Jr., Vector measures, Mathematical Surveys, No. 15, American Mathematical Society, Providence, R.I., 1977. With a foreword by B. J. Pettis. MR 0453964
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.
D. H. Fremlin, The generalized McShane integral, Illinois J. Math. 39 (1995), no. 1, 39–67. MR 1299648
—, Measure theory. Volume 2: Broad foundations, Torres Fremlin, Colchester, 2001.
Russell A. Gordon, The integrals of Lebesgue, Denjoy, Perron, and Henstock, Graduate Studies in Mathematics, vol. 4, American Mathematical Society, Providence, RI, 1994. MR 1288751, DOI 10.1090/gsm/004
Jaroslav Kurzweil and Štefan Schwabik, McShane equi-integrability and Vitali’s convergence theorem, Math. Bohem. 129 (2004), no. 2, 141–157. MR 2073511, DOI 10.14321/realanalexch.29.2.0763
Kazimierz 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
Kazimierz Musiał, Pettis integral, Handbook of measure theory, Vol. I, II, North-Holland, Amsterdam, 2002, pp. 531–586. MR 1954622, DOI 10.1016/B978-044450263-6/50013-0
B. J. Pettis, On integration in vector spaces, Trans. Amer. Math. Soc. 44 (1938), no. 2, 277–304. MR 1501970, DOI 10.1090/S0002-9947-1938-1501970-8
Lawrence H. Riddle and Elias Saab, On functions that are universally Pettis integrable, Illinois J. Math. 29 (1985), no. 3, 509–531. MR 786735
J. Rodríguez, Convergence theorems for the Birkhoff integral, to appear in Houston J. Math., preprint available at http://personales.upv.es/jorodrui.
José Rodríguez, On the existence of Pettis integrable functions which are not Birkhoff integrable, Proc. Amer. Math. Soc. 133 (2005), no. 4, 1157–1163. MR 2117218, DOI 10.1090/S0002-9939-04-07665-8
José Rodríguez, Universal Birkhoff integrability in dual Banach spaces, Quaest. Math. 28 (2005), no. 4, 525–536. MR 2182459, DOI 10.2989/16073600509486145
José Rodríguez, On integration of vector functions with respect to vector measures, Czechoslovak Math. J. 56(131) (2006), no. 3, 805–825. MR 2261655, DOI 10.1007/s10587-006-0058-9
José Rodríguez, The Bourgain property and convex hulls, Math. Nachr. 280 (2007), no. 11, 1302–1309. MR 2337347, DOI 10.1002/mana.200510555
José Rodríguez, Spaces of vector functions that are integrable with respect to vector measures, J. Aust. Math. Soc. 82 (2007), no. 1, 85–109. MR 2301972, DOI 10.1017/S1446788700017481
A. P. Solodov, On the limits of the generalization of the Kolmogorov integral, Mat. Zametki 77 (2005), no. 2, 258–272 (Russian, with Russian summary); English transl., Math. Notes 77 (2005), no. 1-2, 232–245. MR 2157094, DOI 10.1007/s11006-005-0023-1
Michel Talagrand, Pettis integral and measure theory, Mem. Amer. Math. Soc. 51 (1984), no. 307, ix+224. MR 756174, DOI 10.1090/memo/0307