Riemann integration in Banach spaces. (English) Zbl 0764.28008

This paper contains a clear and interesting exposition of several questions related with the Riemann integration of functions mapping (bounded) closed intervals into Banach spaces. One of the main focuses of the paper is to analyse the Banach spaces having the property of Lebesgue (i.e., those spaces having the property that every Riemann integrable function is continuous almost everywhere on the interval). Also it is examined the relationship between the Riemann integral (in Banach spaces) and other (vector-valued) integrals.
Most of the results in this paper are a compilation of works of Graves, Alexiewicz and Orlicz, Rejouani, Nemirovski, Ochan and Rejouani, and da Rocha. Some of the proves presented here are simpler than the original ones.


28B05 Vector-valued set functions, measures and integrals
46G10 Vector-valued measures and integration
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[1] A. Alexiewicz and W. Orlicz, Remarks on Riemann-integration of vector-valued functions , Studia Math. 12 (1951), 125-132. · Zbl 0042.35202
[2] P.G. Casazza and T. Shura, Tsireĺson’s space , Springer-Verlag, Berlin-New York, 1989. · Zbl 0709.46008
[3] J. Diestel and J.J. Uhl, Vector measures , American Mathematical Society, Providence, RI, 1977. · Zbl 0369.46039
[4] L.M. Graves, Riemann integration and Taylor’s theorem in general analysis , Trans. Amer. Math. Soc. 29 (1927), 163-177. JSTOR: · JFM 53.0234.03
[5] R.C. James, Super-reflexive spaces with bases , Pacific J. Math. (2) 41 (1972), 409-419. · Zbl 0235.46031
[6] J. Lindenstrauss and L. Tzafriri, Classical Banach spaces , I, Springer-Verlag, New York, 1977. · Zbl 0362.46013
[7] A.S. Nemirovskii, M.Yu. Ochan, and R. Rejouani, Conditions for Riemann integrability of functions with values in a Banach space , Vestnik Moskov. Univ. Ser. I. Mat. Meh. no. 4, 27 (1972), 62-65.
[8] B.J. Pettis, On integration in vector spaces , Trans. Amer. Math. Soc. 44 (1938), 277-304. JSTOR: · Zbl 0019.41603
[9] R. Rejouani, On the question of the Riemann integrability of functions with values in a Banach space , Vestnik Moskov. Univ. Ser. I Mat. Meh. no. 4 26 (1971), 75-79. · Zbl 0224.46014
[10] ——–, On the question of strong and weak \((R)\)-integrability of abstract functions , Vestnik Moskov. Univ. Ser. I Mat. Meh. no. 6, 26 (1971), 26-31. · Zbl 0228.46035
[11] G.C. da Rocha, Integral de Riemann vetoriel e geometri de espaços de Banach , Ph.D. thesis, Universidade de Sao Paulo, 1979.
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