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Radon-Nikodym property. (English) Zbl 1463.46032

Summary: For a Banach space \(E\) and a probability space \((X,\mathcal{A},\lambda)\), a new proof is given that a measure \(\mu:\mathcal{A}\to E\), with \(\mu\ll\lambda\), has RN derivative with respect to \(\lambda\) iff there is a compact or a weakly compact \(C\subset E\) such that \(|\mu |_{C}:\mathcal{A}\to [0,\infty]\) is a finite valued countably additive measure. Here we define \(|\mu |_{C}(A)=\sup\{\sum_{k}|\langle\mu (A_{k}),f_{k}\rangle |\}\) where \(\{A_{k}\}\) is a finite disjoint collection of elements from \(\mathcal{A}\), each contained in \(A\), and \(\{f_{k}\}\subset E'\) satisfies \(\sup_{k}|f_{k}(C)|\leq 1\). Then the result is extended to the case when \(E\) is a Fréchet space.

MSC:

46B22 Radon-Nikodým, Kreĭn-Milman and related properties
46G10 Vector-valued measures and integration
28B05 Vector-valued set functions, measures and integrals
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References:

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