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

G. Beate Zimmer
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G. Beate Zimmer: Texas A&M University - Corpus Christi, Department of Mathematics and Statistics

Chapter 16 in The Strength of Nonstandard Analysis, 2007, pp 227-237 from Springer

Abstract: 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

Keywords: Banach Space; Bounded Variation; Vector Measure; Generalize Derivative; Nonstandard Analysis (search for similar items in EconPapers)
Date: 2007
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-211-49905-4_16

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DOI: 10.1007/978-3-211-49905-4_16

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