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Universally Lusin-measurable and Baire-$ 1$ projections


Author: Elias Saab
Journal: Proc. Amer. Math. Soc. 78 (1980), 514-518
MSC: Primary 46B22
DOI: https://doi.org/10.1090/S0002-9939-1980-0556623-X
MathSciNet review: 556623
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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.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1980-0556623-X
Keywords: Radon-Nikodým property, universally Lusin-measurable maps
Article copyright: © Copyright 1980 American Mathematical Society

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