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

 

Pettis integrability and the equality of the norms of the weak$ \sp \ast$ integral and the Dunford integral


Author: Elizabeth M. Bator
Journal: Proc. Amer. Math. Soc. 95 (1985), 265-270
MSC: Primary 46G10; Secondary 28B05
MathSciNet review: 801336
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Abstract: If $ (\Omega ,\sum ,\mu )$ is a perfect finite measure space and $ X$ is a Banach space, then it is shown that $ {X^ * }$ has the $ \mu $-Pettis Integral Property if and only if

$\displaystyle \left\Vert {({\text{wea}}{{\text{k}}^ * }) - \int\limits_\Omega {... ...rt = \left\Vert {({\text{Dunford}}) - \int\limits_\Omega {fd\mu } } \right\Vert$

for every bounded weakly measurable function $ f:\Omega \to {X^ * }$.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1985-0801336-4
PII: S 0002-9939(1985)0801336-4
Keywords: Banach space, Pettis integral, Pettis norm, weak measurability
Article copyright: © Copyright 1985 American Mathematical Society