## Universally Lusin-measurable and Baire-$1$ projections

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- by Elias Saab
- Proc. Amer. Math. Soc.
**78**(1980), 514-518 - DOI: https://doi.org/10.1090/S0002-9939-1980-0556623-X
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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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## Bibliographic Information

- © Copyright 1980 American Mathematical Society
- 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