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I am looking for a proof of the Radon–Nikodým 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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    $\begingroup$ One standard reference is Diestel-Uhl. $\endgroup$ – t.b. Nov 2 '11 at 3:35
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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 if one makes appropriate definitions and the relevant research paper is cited.

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