Illinois Journal of Mathematics

A variational Henstock integral characterization of the Radon–Nikodým property

B. Bongiorno, L. Di Piazza, and K. Musiał

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A characterization of Banach spaces possessing the Radon–Nikodým property is given in terms of finitely additive interval functions. We prove that a Banach space $X$ has the RNP if and only if each $X$-valued finitely additive interval function possessing absolutely continuous variational measure is a variational Henstock integral of an $X$-valued function. Due to that characterization several $X$-valued set functions that are only finitely additive can be represented as integrals.

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Illinois J. Math., Volume 53, Number 1 (2009), 87-99.

First available in Project Euclid: 22 January 2010

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Primary: 28B05: Vector-valued set functions, measures and integrals [See also 46G10] 26A39: Denjoy and Perron integrals, other special integrals 46G05: Derivatives [See also 46T20, 58C20, 58C25] 46G10: Vector-valued measures and integration [See also 28Bxx, 46B22] 58C20: Differentiation theory (Gateaux, Fréchet, etc.) [See also 26Exx, 46G05]


Bongiorno, B.; Di Piazza, L.; Musiał, K. A variational Henstock integral characterization of the Radon–Nikodým property. Illinois J. Math. 53 (2009), no. 1, 87--99. doi:10.1215/ijm/1264170840.

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