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Radon-Nikodym theorems for vector valued measures


Author: Joseph Kupka
Journal: Trans. Amer. Math. Soc. 169 (1972), 197-217
MSC: Primary 28A45; Secondary 28A15, 46G10
DOI: https://doi.org/10.1090/S0002-9947-1972-0311871-9
MathSciNet review: 0311871
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Abstract: Let $ \mu $ be a nonnegative measure, and let $ m$ be a measure having values in a real or complex vector space $ V$. This paper presents a comprehensive treatment of the question: When is $ m$ the indefinite integral with respect to $ \mu $ of a $ V$ valued function $ f?$ Previous results are generalized, and two new types of Radon-Nikodym derivative, the ``type $ \rho $'' function and the ``strongly $ \Gamma $ integrable'' function, are introduced. A derivative of type $ \rho $ may be obtained in every previous Radon-Nikodym theorem known to the author, and a preliminary result is presented which gives necessary and sufficient conditions for the measure $ m$ to be the indefinite integral of a type $ \rho $ function. The treatment is elementary throughout, and in particular will include the first elementary proof of the Radon-Nikodym theorem of Phillips.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1972-0311871-9
Keywords: Radon-Nikodym theorems, vector valued measures, the lifting theorem, decomposable measure spaces
Article copyright: © Copyright 1972 American Mathematical Society

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