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Proceedings of the American Mathematical Society

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

 

 

The Jordan decomposition of vector-valued measures


Authors: B. Faires and T. J. Morrison
Journal: Proc. Amer. Math. Soc. 60 (1976), 139-143
MSC: Primary 28A45; Secondary 46G10
DOI: https://doi.org/10.1090/S0002-9939-1976-0419723-X
MathSciNet review: 0419723
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Abstract: This paper gives criteria for a vector-valued Jordan decomposition theorem to hold. In particular, suppose L is an order complete vector lattice and $ \mathcal{A}$ is a Boolean algebra. Then an additive set function $ \mu :\mathcal{A} \to L$ can be expressed as the difference of two positive additive measures if and only if $ \mu (\mathcal{A})$ is order bounded. A sufficient condition for a countably additive set function with values in $ {c_0}(\Gamma )$, for any set $ \Gamma $, to be decomposed into difference of countably additive set functions is given; namely, the domain being the power set of some set.


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DOI: https://doi.org/10.1090/S0002-9939-1976-0419723-X
Article copyright: © Copyright 1976 American Mathematical Society