Czechoslovak Mathematical Journal, first online, pp. 1-9


A remark on weak McShane integral

Kazushi Yoshitomi

Received March 31, 2017.   Published online August 6, 2018.

Abstract:  We characterize the weak McShane integrability of a vector-valued function on a finite Radon measure space by means of only finite McShane partitions. We also obtain a similar characterization for the Fremlin generalized McShane integral.
Keywords:  weak McShane integral; finite McShane partition; Radon measure space
Classification MSC:  28B05
DOI:  10.21136/CMJ.2018.0153-17

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References:
[1] H. Bauer: Measure and Integration Theory. De Gruyter Studies in Mathematics 26. Walter de Gruyter, Berlin (2001). MR 1897176 | Zbl 0985.28001
[2] C.-A. Faure, J. Mawhin: Integration over unbounded multidimensional intervals. J. Math. Anal. Appl. 205 (1997), 65-77. DOI 10.1006/jmaa.1996.5172 | MR 1426980 | Zbl 0879.26047
[3] D. H. Fremlin: Topological Riesz Spaces and Measure Theory. Cambridge University Press, London (1974). MR 0454575 | Zbl 0273.46035
[4] D. H. Fremlin: The generalized McShane integral. Ill. J. Math. 39 (1995), 39-67. MR 1299648 | Zbl 0810.28006
[5] J. L. Kelley, T. P. Srinivasan: Measure and Integral Vol. 1. Graduate Texts in Mathematics 116. Springer, New York (1988). DOI 10.1007/978-1-4612-4570-4 | MR 0918770 | Zbl 0635.28001
[6] M. Saadoune, R. Sayyad: The weak McShane integral. Czech. Math. J. 64 (2014), 387-418. DOI 10.1007/s10587-014-0108-7 | MR 3277743 | Zbl 1340.28016
[7] Š. Schwabik, G. Ye: Topics in Banach Space Integration. Series in Real Analysis Vol. 10. World Scientific, Singapore (2005). DOI 10.1142/9789812703286 | MR 2167754 | Zbl 1088.28008
[8] M. Talagrand: Pettis integral and measure theory. Mem. Am. Math. Soc. 307 (1984). DOI 10.1090/memo/0307 | MR 0756174  | Zbl 0582.46049

Affiliations:   Kazushi Yoshitomi, Department of Mathematics and Information Sciences, Tokyo Metropolitan University, Minamiohsawa 1-1, Hachioji, Tokyo 192-0397, Japan, e-mail: yositomi@tmu.ac.jp


 
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