Radon-Nikodym theorems for vector valued measures

Kupka, Joseph

doi:10.1090/S0002-9947-1972-0311871-9

Radon-Nikodym theorems for vector valued measures
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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.

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