zbMATH — the first resource for mathematics

The weak McShane integral. (English) Zbl 1340.28016
The authors introduce a modification of the generalized McShane integration process due to D. H. Fremlin [Ill. J. Math. 39, No. 1, 39–67 (1995; Zbl 0810.28006)] that involves Banach space-valued integrands on the \(\sigma \)-finite outer regular quasi Radon measure space \((S,\Sigma,\mathcal {T},\mu)\). This new integral, the weak McShane integral in the author’s terminology, is defined in the same manner as the Fremlin integral to be the gauge limit of the generalized McShane sums in the weak topology, rather than in the norm topology.
The key results of the paper under review compare the weak McShane integral with the Pettis integral:
A function is weakly McShane integrable on each measurable subset of \(S\) if and only if it is both Pettis and weakly McShane integrable on \(S\).
If a function is weakly McShane integrable on \(S\), then there exist measurable sets of finite measure \((S_{l})_{l\geq 1}\) that cover \(S\) on each of which the function is Pettis integrable.
If the range does not contain an isomorphic copy of \(c_{0}\), then each weakly McShane integrable function is necessarily Pettis integrable. Moreover, the assumption that the range does not contain an isomorphic copy of \(c_{0}\) cannot be removed from this theorem.

28B05 Vector-valued set functions, measures and integrals
46G10 Vector-valued measures and integration
26A39 Denjoy and Perron integrals, other special integrals
Full Text: DOI
[1] A. Aizpuru, F. J. Pérez-Fernández: Characterizations of series in Banach spaces. Acta Math. Univ. Comen., New Ser. 68 (1999), 337–344. · Zbl 0952.46009
[2] C. Castaing: Weak compactness and convergence in Bochner and Pettis integration. Vietnam J. Math. 24 (1996), 241–286.
[3] R. Deville, J. Rodríguez: Integration in Hilbert generated Banach spaces. Isr. J. Math. 177 (2010), 285–306. · Zbl 1216.46016
[4] J. Diestel, J. J. Uhl Jr.: Vector Measures. Mathematical Surveys 15, AMS, Providence, R. I., 1977. · Zbl 0369.46039
[5] L. Di Piazza, D. Preiss: When do McShane and Pettis integrals coincide? Ill. J. Math. 47 (2003), 1177–1187. · Zbl 1045.28006
[6] M. Fabian, G. Godefroy, P. Hájek, V. Zizler: Hilbert-generated spaces. J. Funct. Anal. 200 (2003), 301–323. · Zbl 1039.46015
[7] D. H. Fremlin: The generalized McShane integral. Ill. J. Math. 39 (1995), 39–67. · Zbl 0810.28006
[8] D. H. Fremlin: Measure Theory. Vol. 2. Broad Foundations. Corrected second printing of the 2001 original, Torres Fremlin, Colchester, 2003.
[9] D. H. Fremlin: Measure theory. Vol. 4. Topological Measure Spaces. Part I, II. Corrected second printing of the 2003 original, Torres Fremlin, Colchester, 2006.
[10] D. H. Fremlin, J. Mendoza: On the integration of vector-valued functions. Ill. J. Math. 38 (1994), 127–147. · Zbl 0790.28004
[11] R. F. Geitz: Pettis integration. Proc. Am. Math. Soc. 82 (1981), 81–86. · Zbl 0506.28007
[12] R. A. Gordon: The McShane integral of Banach-valued functions. Ill. J. Math. 34 (1990), 557–567. · Zbl 0685.28003
[13] K. Musiał: Vitali and Lebesgue convergence theorems for Pettis integral in locally convex spaces. Atti Semin. Math. Fis. Univ. Modena 35 (1987), 159–165. · Zbl 0636.28005
[14] J. Rodríguez: On the equivalence of McShane and Pettis integrability in non-separable Banach spaces. J. Math. Anal. Appl. 341 (2008), 80–90. · Zbl 1138.28003
[15] M. Saadoune, R. Sayyad: From scalar McShane integrability to Pettis integrability. Real Anal. Exchange 38 (2012–2013), 445–466.
[16] Š. Schwabik, G. Ye: Topics in Banach Space Integration. Series in Real Analysis 10, World Scientific, Hackensack, 2005. · Zbl 1088.28008
[17] G. Ye, Š. Schwabik: The McShane and the weak McShane integrals of Banach spacevalued functions defined on \(\mathbb{R}\)m. Math. Notes, Miskolc 2 (2001), 127–136. · Zbl 0993.28005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.