A characterization of Banach spaces possessing the Radon-Nikodym 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.
Bongiorno, B., Di Piazza, L., Musial, K. (2009). A variational Henstock integral characterization of the Radon-Nikodym property. ILLINOIS JOURNAL OF MATHEMATICS, 53(1), 87-99.
A variational Henstock integral characterization of the Radon-Nikodym property
BONGIORNO, Benedetto;DI PIAZZA, Luisa;
2009-01-01
Abstract
A characterization of Banach spaces possessing the Radon-Nikodym 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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