Radon-Nikodym theorems for vector valued measures

Abstract

Let μ \mu be a nonnegative measure, and let m m be a measure having values in a real or complex vector space V V . This paper presents a comprehensive treatment of the question: When is m m the indefinite integral with respect to μ \mu of a V V valued function f ? f? Previous results are generalized, and two new types of Radon-Nikodym derivative, the “type ρ \rho ” function and the “strongly Γ \Gamma integrable” function, are introduced. A derivative of type ρ \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 m to be the indefinite integral of a type ρ \rho function. The treatment is elementary throughout, and in particular will include the first elementary proof of the Radon-Nikodym theorem of Phillips.