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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?)

  • [1] R. G. Bartle, N. Dunford, and J. Schwartz, Weak compactness and vector measures, Canad. J. Math. 7 (1955), 289–305. MR 0070050
  • [2] J. Diestel and J. J. Uhl Jr., Vector measures, American Mathematical Society, Providence, R.I., 1977. With a foreword by B. J. Pettis; Mathematical Surveys, No. 15. MR 0453964
  • [3] N. Dunford and J. T. Schwartz, Linear operators, Part I, Interscience, New York, 1958.
  • [4] S. Mazur, Über konvexe Mengen in linearen normierte Räumen, Studia Math. 4 (1933), 70-84.
  • [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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Article copyright: © Copyright 1982 American Mathematical Society