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)



Universally Lusin-measurable and Baire-$ 1$ projections

Author: Elias Saab
Journal: Proc. Amer. Math. Soc. 78 (1980), 514-518
MSC: Primary 46B22
MathSciNet review: 556623
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: It is obvious that a dual Banach space $ {E^\ast}$ is reflexive if and only if the natural projection P from $ {E^{ \ast \ast \ast }}$ to $ {E^\ast}$ is $ {\text{weak}^\ast}$ to weak continuous. In this paper it is proved that the next best condition on P, namely that P is $ {\text{weak}^\ast}$ to weak universally Lusin-measurable is necessary and sufficient for $ {E^\ast}$ to have the Radon-Nikodým property. In addition we prove that if E is any Banach space that is complemented in its second dual by a $ {\text{weak}^\ast}$ to weak Baire-1 projection, then E has the Radon-Nikodým property. We also prove that if E is a Banach space that is complemented in its second dual $ {E^{ \ast \ast }}$ by a projection $ P:{E^{\ast \ast}} \to E$ with $ F = {P^{ - 1}}(0)$ weakly K-analytic; then saying that $ {E^{ \ast \ast }}$ has the Radon-Nikodým property is equivalent to saying that P is $ {\text{weak}^\ast}$ to weak universally Lusin-measurable.

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

  • [1] A. Badrikian, Séminaire sur les fonctions aléatoires linéaires et les mesures cylindriques, Lectures Notes in Math., vol. 139, Springer-Verlag, Berlin and New York, 1970. MR 0279271 (43:4994)
  • [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. Dinculeanu, Vector measures, Vebdeutscher Verlag der Wisenschaften, Berlin, 1966. MR 0206189 (34:6011a)
  • [4] R.C. James, A separable somewhat reflexive Banach space with nonseparable dual, Bull. Amer. Math. Soc. 80 (1974), 738-743. MR 0417763 (54:5811)
  • [5] T. Kuo, On conjugate Banach spaces with the Radon-Nikodým property, Pacific J. Math. 59, 497-503. MR 0394133 (52:14938)
  • [6] E. Odell and H.P. Rosenthal, A double dual characterization of separable Banach spaces containing $ {l_1}$, Israel J. Math. 20 (1975), 375-384. MR 0377482 (51:13654)
  • [7] E. Saab, Une charactérisation des convexes $ \sigma (E',E)$ compact possédant la proprieté de Radon-Nikodým, C.R. Acad. Sci. Paris Sér. A-B 286 (1978), 45-48. MR 0511756 (58:23502)
  • [8] -, The Radon-Nikodým property, weak K-analyticity and universal measurability, Ph.D. Dissertation, Univ. of Illinois, 1978-1979.
  • [9] L. Schwartz, Radon measures on arbitrary topological spaces and cylindrical measures, Tata Institute, Oxford Univ. Press, Oxford, 1973. MR 0426084 (54:14030)
  • [10] -, Propriété de Radon-Nikodým, Séminaire Maurey-Schwartz (1974-1975), Centre Math., École Polytech., Paris, 1975, Exp. No. V-VI.
  • [11] M. Talagrand, Sur une conjecture de H.H. Corson, Bull. Sci. Math. (2) 99 (1975), 211-212. MR 0430752 (55:3757)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 46B22

Retrieve articles in all journals with MSC: 46B22

Additional Information

Keywords: Radon-Nikodým property, universally Lusin-measurable maps
Article copyright: © Copyright 1980 American Mathematical Society

American Mathematical Society