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Proceedings of the American Mathematical Society

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

 

 

Vector measure Banach spaces containing a complemented copy of $c_{0}$


Authors: A. Picón and C. Piñeiro
Journal: Proc. Amer. Math. Soc. 132 (2004), 2893-2898
MSC (2000): Primary 46G10
DOI: https://doi.org/10.1090/S0002-9939-04-07518-5
Published electronically: May 21, 2004
MathSciNet review: 2063108
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Abstract: Let $ X $ a Banach space and $ \Sigma $ a $\sigma$-algebra of subsets of a set $\Omega $. We say that a vector measure Banach space $ (\mathcal{M} (\Sigma , X ) , \Vert \cdot \Vert _\mathcal{M })$has the bounded Vitaly-Hahn-Sacks Property if it satisfies the following condition: Every vector measure $ m : \Sigma \longrightarrow X $ , for which there exists a bounded sequence $(m_{n})$ in $\mathcal{M } (\Sigma, X )$ verifying $ \displaystyle\lim_{n \to \infty} m_{n} ( A ) = m(A)$ for all $A \in \Sigma $, must belong to $\mathcal{M} (\Sigma, X)$. Among other results, we prove that, if $\mathcal{M}(\Sigma, X)$ is a vector measure Banach space with the bounded V-H-S Property and containing a complemented copy of $c_{0}$, then $X$ contains a copy of $c_{0}$.


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Additional Information

A. Picón
Affiliation: Departamento de Matemáticas, Facultad de Ciencias Experimentales, Campus universitario de El Carmen, Universidad de Huelva, 21071, Huelva, Spain

C. Piñeiro
Affiliation: Departamento de Matemáticas, Facultad de Ciencias Experimentales, Campus universitario de El Carmen, Universidad de Huelva, 21071, Huelva, Spain
Email: candido@uhu.es

DOI: https://doi.org/10.1090/S0002-9939-04-07518-5
Received by editor(s): May 14, 2002
Published electronically: May 21, 2004
Communicated by: Jonathan M. Borwein
Article copyright: © Copyright 2004 American Mathematical Society