A Radon-Nikodým theorem for a vector-valued reference measure

Abstract

The conclusion of a Radon-Nikodým theorem is that a measure μ can be represented as an integral with respect to a reference measure such that for all measurable sets A, μ(A) = ∫_A f _μ(x)dλ with a (Bochner or Lebesgue) integrable derivative or density f _μ. The measure λ is usually a countably additive σ-finite measure on the given measure space and the measure μ is absolutely continuous with respect to λ. Different theorems have different range spaces for μ. which could be the real numbers, or Banach spaces with or without the Radon-Nikodým property. In this paper we generalize to derivatives of vector valued measures with respect a vector-valued reference measure. We present a Radon-Nikodým theorem for vector measures of bounded variation that are absolutely continuous with respect to another vector measure of bounded variation. While it is easy in settings such as μ << λ, where λ is Lebesgue measure on the interval [0,1] and μ is vector-valued to write down a nonstandard Radon-Nikodým derivative of the form ϕ : *[0,1] → fin(*E) by \( \varphi _\mu (x) = \sum\nolimits_{i = 1}^H {\tfrac{{{}^*\mu (A_i )}} {{{}^*\lambda (A_i )}}1_{A_i } (x)} \) a vector valued reference measure does not allow this approach, as the quotient of two vectors in different Banach spaces is undefined. Furthermore, generalizing to a vector valued control measure necessitates the use of a generalization of the Bartle integral, a bilinear vector integral.