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Differentiability of convex functions and Rybakov's theorem


Authors: Russell G. Bilyeu and Paul W. Lewis
Journal: Proc. Amer. Math. Soc. 86 (1982), 186-187
MSC: Primary 28B05; Secondary 46G10
MathSciNet review: 663894
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Abstract: Rybakov proved that if $ \mu :\Sigma \to X$ is a countably additive Banach space valued measure on a $ \sigma $-algebra $ \Sigma $, then there is an element $ {x^ * } \in {X^ * }$ so that $ \mu \ll {x^ * }\mu $. In this note, we show that Rybakov's theorem follows essentially from a classical result of Mazur on the Gateaux differentiability of convex functions.


References [Enhancements On Off] (What's this?)

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  • [5] V. I. Rybakov, On the theorem of Bartle, Dunford and Schwartz on vector-valued measures, Mat. Zametki 7 (1970), 247–254 (Russian). MR 0260971

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DOI: https://doi.org/10.1090/S0002-9939-1982-0663894-X
Article copyright: © Copyright 1982 American Mathematical Society