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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Set-valued measures


Author: Zvi Artstein
Journal: Trans. Amer. Math. Soc. 165 (1972), 103-125
MSC: Primary 28A45
DOI: https://doi.org/10.1090/S0002-9947-1972-0293054-4
MathSciNet review: 0293054
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Abstract: A set-valued measure is a $ \sigma $-additive set-function which takes on values in the nonempty subsets of a euclidean space. It is shown that a bounded and non-atomic set-valued measure has convex values. Also the existence of selectors (vector-valued measures) is investigated. The Radon-Nikodym derivative of a set-valued measure is a set-valued function. A general theorem on the existence of R.-N. derivatives is established. The techniques require investigations of measurable set-valued functions and their support functions.


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DOI: https://doi.org/10.1090/S0002-9947-1972-0293054-4
Keywords: Set-valued measure, measurable set-valued function, support function, integral of a set-valued function, selectors, Lyapunov's convexity theorem, Radon-Nikodym derivatives
Article copyright: © Copyright 1972 American Mathematical Society