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

In: The Strength of Nonstandard Analysis

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  • G. Beate Zimmer

    (Texas A&M University - Corpus Christi, Department of Mathematics and Statistics)

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

Suggested Citation

  • G. Beate Zimmer, 2007. "A Radon-Nikodým theorem for a vector-valued reference measure," Springer Books, in: Imme van den Berg & Vítor Neves (ed.), The Strength of Nonstandard Analysis, chapter 16, pages 227-237, Springer.
    • Handle: RePEc:spr:sprchp:978-3-211-49905-4_16
    DOI: 10.1007/978-3-211-49905-4_16
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