Abstract
In general, Banach space-valued Riemann integrable functions defined on [0, 1] (equipped with the Lebesgue measure) need not be weakly continuous almost everywhere. A Banach space is said to have the weak Lebesgue property if every Riemann integrable function taking values in it is weakly continuous almost everywhere. In this paper we discuss this property for the Banach space L 1 X of all Bochner integrable functions from [0, 1] to the Banach space X. We show that L 1 X has the weak Lebesgue property whenever X has the Radon-Nikodým property and X* is separable. This generalizes the result by Chonghu Wang and Kang Wan [Rocky Mountain J. Math., 31(2), 697–703 (2001)] that L 1[0, 1] has the weak Lebesgue property.
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The first author is supported by MEC and FEDER (Project MTM2005-08350-C03-03) and Generalitat Valenciana (Project GV/2007/191); the second author is supported by MEC and FEDER (Project MTM2005-08379), Fundación Séneca (Project 00690/PI/04) and the “Juan de la Cierva” Programme (MEC and FSE); the third author is supported by MEC and FEDER (Project MTM2006-11690-C02-01)
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Calabuig, J.M., Rodríguez, J. & Sánchez-Pérez, E.A. Weak continuity of Riemann integrable functions in Lebesgue-Bochner spaces. Acta. Math. Sin.-English Ser. 26, 241–248 (2010). https://doi.org/10.1007/s10114-010-7382-6
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DOI: https://doi.org/10.1007/s10114-010-7382-6