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.


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


[1] Davis W. J.; Figiel T.; Johnson W. B.; Pelczynski A., Factoring weakly compact operators, J. Funct. Anal. 17 (1974), 311-327 · Zbl 0306.46020
[2] Diestel J.; Uhl J. J., Vector Measures, Amer. Math. Soc. Surveys, 15, American Mathematical Society, Providence, RI, 1977 · Zbl 0521.46035
[3] Gruenwald M. E.; Wheeler R. F., A strict representation of \(L_1(\mu, X)\), J. Math. Anal. Appl. 155 (1991), 140-155 · Zbl 0736.46027
[4] Khurana S. S., Topologies on spaces of continuous vector-valued functions, Trans Amer. Math. Soc. 241 (1978), 195-211 · Zbl 0335.46017
[5] Khurana S. S., Topologies on spaces of continuous vector-valued functions II, Math. Ann. 234 (1978), 159-166 · Zbl 0362.46035
[6] Khurana S. S., Pointwise compactness and measurability, Pacific J. Math. 83 (1979), 387-391 · Zbl 0425.46009
[7] Phelps R. R., Lectures on Choquet’s Theorem, D. van Nostrand Company, Inc., Princeton, N.J.-Toronto, Ont.-London, 1966 · Zbl 0997.46005
[8] Schaefer H. H., Topological Vector Spaces, Springer, 1986 · Zbl 0983.46002
[9] Ionescu Tulcea A.; Ionescu Tulcea C., Topics in the theory of lifting, Springer, New York, 1969 · Zbl 0179.46303
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.