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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Vector valued functions equivalent to measurable functions


Author: J. J. Uhl
Journal: Proc. Amer. Math. Soc. 68 (1978), 32-36
MSC: Primary 28A20
MathSciNet review: 0466473
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Abstract: Let X be a Banach space with dual $ {X^\ast}$ and let $ (\Omega ,\Sigma ,\mu )$ be a finite measure space. Suppose $ f:\Omega \to X$ is weakly measurable. There exists a (norm) measurable $ g:\Omega \to X$ such that $ \langle {x^\ast},f\rangle = \langle {x^\ast},g\rangle $ a.e. for each $ {x^\ast} \in {X^\ast}$ if and only if each set A of positive $ \mu $-measure has a subset B of positive $ \mu $-measure such that there is a weakly compact convex subset K of X with the property that

$\displaystyle \langle {x^\ast},f\rangle \leqslant \mathop {\sup }\limits_{x \in K} \;\langle {x^\ast},x\rangle $

$ \mu $-almost everywhere on B for each $ {x^\ast} \in {X^\ast}$.

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DOI: http://dx.doi.org/10.1090/S0002-9939-1978-0466473-1
PII: S 0002-9939(1978)0466473-1
Article copyright: © Copyright 1978 American Mathematical Society