Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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
DOI: https://doi.org/10.1090/S0002-9939-1982-0663894-X
MathSciNet review: 663894
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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. T. Schwartz, Weak compactness and vector measures, Canad. J. Math. 7 (1955), 289-305. MR 0070050 (16:1123c)
  • [2] J. Diestel and J. J. Uhl, Jr., Vector measures, Math. Surveys, no. 15, Amer. Math. Soc., Providence, R. I., 1977. MR 0453964 (56:12216)
  • [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. Rybakov, Theorem of Bartle, Dunford, and Schwartz on vector-valued measures, Mat. Zametki 7 (1970), 247-254. MR 0260971 (41:5591)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 28B05, 46G10

Retrieve articles in all journals with MSC: 28B05, 46G10


Additional Information

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

American Mathematical Society