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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Radon-Nikodym theorems for vector valued measures
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by Joseph Kupka PDF
Trans. Amer. Math. Soc. 169 (1972), 197-217 Request permission

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.
References
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Additional Information
  • © Copyright 1972 American Mathematical Society
  • 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