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The Radon-Nikodym property and dentable sets in Banach spaces


Authors: W. J. Davis and R. R. Phelps
Journal: Proc. Amer. Math. Soc. 45 (1974), 119-122
MSC: Primary 46B05
DOI: https://doi.org/10.1090/S0002-9939-1974-0344852-7
MathSciNet review: 0344852
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Abstract: In order to prove a Radon-Nikodym theorem for the Bochner integral, Rieffel [5] introduced the class of ``dentable'' subsets of Banach spaces. Maynard [3] later introduced the strictly larger class of ``$ s$-dentable'' sets, and extended Rieffel's result to show that a Banach space has the Radon-Nikodym property if and only if every bounded nonempty subset of $ E$ is $ s$-dentable. He left open, however, the question as to whether, in a space with the Radon-Nikodym property, every bounded nonempty set is dentable. In the present note we give an elementary construction which shows this question has an affirmative answer.


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DOI: https://doi.org/10.1090/S0002-9939-1974-0344852-7
Keywords: Banach space, dentable sets, vector valued measures, Radon-Nikodym theorem
Article copyright: © Copyright 1974 American Mathematical Society