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I am looking for a proof of the Radon-Nikodym theorem in the case of vector valued measures. Many textbooks cover the scalar case. The book I am reading mentions the vector valued case but does not provide a proof or a reference. Any help is greatly appreciated.

Thanks, Phanindra

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One standard reference is Diestel-Uhl. –  t.b. Nov 2 '11 at 3:35

2 Answers 2

The Radon-Nikodym theorem is false in general for Banach valued measures. Banach spaces for which the RN theorem theorem holds are said to have RNP (Radon-Nikodym Property). For example, Hilbert spaces, reflexive space. On the other hand, $L^1$ does not have RNP. You may check the Wikipedia article.

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radon nikodym theorem holds for hilbert valued measures. a proof can be found in analysis II serge lang Addison Welsly 1968 or real and functional analysis 3rd edition serge lang springer verlag 1993. there it is stated that a version of radon nikodym theprem can hold for banach spaces with appropriate definitions and relevant research paper is cited.

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