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Title: Radon-Nikodym property (English)
Author: Khurana, Surjit Singh
Language: English
Journal: Commentationes Mathematicae Universitatis Carolinae
ISSN: 0010-2628 (print)
ISSN: 1213-7243 (online)
Volume: 58
Issue: 4
Year: 2017
Pages: 461-464
Summary lang: English
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Category: math
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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 Frechet space. (English)
Keyword: liftings
Keyword: lifting topology
Keyword: weakly compact sets
Keyword: Radon-Nikodym derivative
MSC: 28A51
MSC: 28B05
MSC: 28C05
MSC: 46B22
MSC: 46G05
MSC: 46G10
MSC: 60B05
idZBL: Zbl 06837079
idMR: MR3737118
DOI: 10.14712/1213-7243.2015.228
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Date available: 2017-12-12T06:46:38Z
Last updated: 2020-01-05
Stable URL: http://hdl.handle.net/10338.dmlcz/146990
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Reference: [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, MR 0355536, 10.1016/0022-1236(74)90044-5
Reference: [2] Diestel J., Uhl J.J.: Vector Measures.Amer. Math. Soc. Surveys, 15, American Mathematical Society, Providence, RI, 1977. Zbl 0521.46035, MR 0453964
Reference: [3] Gruenwald M.E., Wheeler R.F.: A strict representation of $L_{1}(\mu, X)$.J. Math. Anal. Appl. 155 (1991), 140–155. MR 1089331
Reference: [4] Khurana S.S.: Topologies on spaces of continuous vector-valued functions.Trans Amer. Math. Soc. 241 (1978), 195–211. MR 0492297, 10.1090/S0002-9947-1978-0492297-X
Reference: [5] Khurana S.S.: Topologies on spaces of continuous vector-valued functions II.Math. Ann. 234 (1978), 159–166. MR 0494178, 10.1007/BF01420966
Reference: [6] Khurana S.S.: Pointwise compactness and measurability.Pacific J. Math. 83 (1979), 387–391. Zbl 0425.46009, MR 0557940, 10.2140/pjm.1979.83.387
Reference: [7] Phelps R.R.: Lectures on Choquet's Theorem.D. van Nostrand Company, Inc., Princeton, N.J.-Toronto, Ont.-London, 1966. Zbl 0997.46005, MR 0193470
Reference: [8] Schaefer H.H.: Topological Vector Spaces.Springer, 1986. Zbl 0983.46002, MR 0342978
Reference: [9] Ionescu Tulcea A., Ionescu Tulcea C.: Topics in the theory of lifting.Springer, New York, 1969. Zbl 0179.46303, MR 0276438
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