Radon-Nikodym property

Surjit Singh Khurana

Commentationes Mathematicae Universitatis Carolinae (2017)

  • Volume: 58, Issue: 4, page 461-464
  • ISSN: 0010-2628

Abstract

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For a Banach space E and a probability space ( X , 𝒜 , λ ) , a new proof is given that a measure μ : 𝒜 → E , with μ ≪ λ , has RN derivative with respect to λ iff there is a compact or a weakly compact C ⊂ E such that | μ | C : 𝒜 → [ 0 , ∞ ] is a finite valued countably additive measure. Here we define | μ | C ( A ) = sup { ∑ k | 〈 μ ( A k ) , f k 〉 | } where { A k } is a finite disjoint collection of elements from 𝒜 , each contained in A , and { f k } ⊂ E ' satisfies sup k | f k ( C ) | ≤ 1 . Then the result is extended to the case when E is a Frechet space.

How to cite

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Khurana, Surjit Singh. "Radon-Nikodym property." Commentationes Mathematicae Universitatis Carolinae 58.4 (2017): 461-464. <http://eudml.org/doc/294585>.

@article{Khurana2017,
abstract = {For a Banach space $E$ and a probability space $(X, \mathcal \{A\}, \lambda )$, a new proof is given that a measure $\mu : \mathcal \{A\} \rightarrow E$, 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 $|\mu |_\{C\} : \mathcal \{A\} \rightarrow [0, \infty ]$ is a finite valued countably additive measure. Here we define $|\mu |_\{C\}(A) = \sup \lbrace \sum _\{k\} |\langle \mu (A_\{k\}), f_\{k\}\rangle |\rbrace $ where $\lbrace A_\{k\}\rbrace $ is a finite disjoint collection of elements from $\mathcal \{A\}$, each contained in $A$, and $\lbrace f_\{k\}\rbrace \subset E^\{\prime \}$ satisfies $\sup _\{k\} |f_\{k\} (C)|\le 1$. Then the result is extended to the case when $E$ is a Frechet space.},
author = {Khurana, Surjit Singh},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {liftings; lifting topology; weakly compact sets; Radon-Nikodym derivative},
language = {eng},
number = {4},
pages = {461-464},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Radon-Nikodym property},
url = {http://eudml.org/doc/294585},
volume = {58},
year = {2017},
}

TY - JOUR
AU - Khurana, Surjit Singh
TI - Radon-Nikodym property
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2017
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 58
IS - 4
SP - 461
EP - 464
AB - For a Banach space $E$ and a probability space $(X, \mathcal {A}, \lambda )$, a new proof is given that a measure $\mu : \mathcal {A} \rightarrow E$, 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 $|\mu |_{C} : \mathcal {A} \rightarrow [0, \infty ]$ is a finite valued countably additive measure. Here we define $|\mu |_{C}(A) = \sup \lbrace \sum _{k} |\langle \mu (A_{k}), f_{k}\rangle |\rbrace $ where $\lbrace A_{k}\rbrace $ is a finite disjoint collection of elements from $\mathcal {A}$, each contained in $A$, and $\lbrace f_{k}\rbrace \subset E^{\prime }$ satisfies $\sup _{k} |f_{k} (C)|\le 1$. Then the result is extended to the case when $E$ is a Frechet space.
LA - eng
KW - liftings; lifting topology; weakly compact sets; Radon-Nikodym derivative
UR - http://eudml.org/doc/294585
ER -

References

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  5. Khurana S.S., 10.1007/BF01420966, Math. Ann. 234 (1978), 159–166. MR0494178DOI10.1007/BF01420966
  6. Khurana S.S., 10.2140/pjm.1979.83.387, Pacific J. Math. 83 (1979), 387–391. Zbl0425.46009MR0557940DOI10.2140/pjm.1979.83.387
  7. Phelps R.R., Lectures on Choquet's Theorem, D. van Nostrand Company, Inc., Princeton, N.J.-Toronto, Ont.-London, 1966. Zbl0997.46005MR0193470
  8. Schaefer H.H., Topological Vector Spaces, Springer, 1986. Zbl0983.46002MR0342978
  9. Ionescu Tulcea A., Ionescu Tulcea C., Topics in the theory of lifting, Springer, New York, 1969. Zbl0179.46303MR0276438

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