# Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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## Universally Lusin-measurable and Baire-$1$ projectionsHTML articles powered by AMS MathViewer

by Elias Saab
Proc. Amer. Math. Soc. 78 (1980), 514-518 Request permission

## 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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