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

Rodríguez, José

doi:10.1090/S0002-9939-04-07665-8

On the existence of Pettis integrable functions which are not Birkhoff integrable
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by José Rodríguez PDF

Proc. Amer. Math. Soc. 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

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, The Birkhoff integral and the property of Bourgain, To appear in Math. Ann.
Donald L. Cohn, Measure theory, Birkhäuser Boston, Inc., Boston, MA, 1993. Reprint of the 1980 original. MR 1454121
L. Di Piazza and D. Preiss, When do McShane and Pettis integrals coincide?, Illinois J. Math. 47 (2003), no. 4, 1177–1187. MR 2036997, DOI 10.1215/ijm/1258138098
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, DOI 10.1090/surv/015
Marián J. 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
Marián Fabian, Petr Habala, Petr Hájek, Vicente Montesinos Santalucía, Jan Pelant, and Václav 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, DOI 10.1007/978-1-4757-3480-5
D. H. Fremlin, The generalized McShane integral, Illinois J. Math. 39 (1995), no. 1, 39–67. MR 1299648, DOI 10.1215/ijm/1255986628
—, The McShane and Birkhoff integrals of vector-valued functions, University of Essex Mathematics Department Research Report 92-10, 1999.
D. H. Fremlin and J. Mendoza, On the integration of vector-valued functions, Illinois J. Math. 38 (1994), no. 1, 127–147. MR 1245838, DOI 10.1215/ijm/1255986891
Russell Gordon, Riemann integration in Banach spaces, Rocky Mountain J. Math. 21 (1991), no. 3, 923–949. MR 1138145, DOI 10.1216/rmjm/1181072923
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 (English, with English, Russian and Ukrainian summaries). MR 1760946
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
R. S. Phillips, Integration in a convex linear topological space, Trans. Amer. Math. Soc. 47 (1940), 114–145. MR 2707, DOI 10.1090/S0002-9947-1940-0002707-3
A. N. Plichko, On projective resolutions of the identity operator and Markushevich bases, Soviet Math. Dokl. 25 (1982), no. 2, 386–389.
Michel Talagrand, Pettis integral and measure theory, Mem. Amer. Math. Soc. 51 (1984), no. 307, ix+224. MR 756174, DOI 10.1090/memo/0307
M. Valdivia, Simultaneous resolutions of the identity operator in normed spaces, Collect. Math. 42 (1991), no. 3, 265–284 (1992). MR 1203185