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On the Sobczyk-Hammer decomposition of additive set functions


Author: Wilfried Siebe
Journal: Proc. Amer. Math. Soc. 86 (1982), 447-450
MSC: Primary 28A10
DOI: https://doi.org/10.1090/S0002-9939-1982-0671212-6
MathSciNet review: 671212
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Abstract: It is observed that continuity for charges is equivalent to the absence of two-valued minorants. This characterization forms the basis of a new short proof within a functional-analytic context of a decomposition theorem by A. Sobczyk and P. C. Hammer [5] for charges on a field $ \mathfrak{A}$ into a continuous part and a part which can be written as a sum of at most two-valued charges on $ \mathfrak{A}$. A counterexample shows that in general the decomposition of a charge into a nonatomic part and a part which has no nonnull nonatomic minorant is not unique.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1982-0671212-6
Keywords: Continuous resp. nonatomic charges, Krein-Milman representation theorem, decomposition theorem
Article copyright: © Copyright 1982 American Mathematical Society