scholarly journals A variational Henstock integral characterization of the Radon–Nikodým property

2009 ◽  
Vol 53 (1) ◽  
pp. 87-99 ◽  
Author(s):  
B. Bongiorno ◽  
L. Di Piazza ◽  
K. Musiał
Keyword(s):  
Henstock Integral ◽  
Nikodym Property

A variational mcshane integral characterization of the Radon-Nikodym property

2013 ◽  
Vol 63 (3) ◽  
Author(s):  
Sokol Kaliaj
Keyword(s):  
Banach Spaces ◽  
Lebesgue Measure ◽  
Absolutely Continuous ◽  
Mcshane Integral ◽  
Nikodym Property ◽  
Interval Functions

AbstractWe present a new characterization of Banach spaces possessing the Radon-Nikodym property in terms of additive interval functions whose McShane variational measures are absolutely continuous with respect to the Lebesgue measure.

scholarly journals A characterization of the Radon–Nikodym property

2013 ◽  
Vol 405 (1) ◽  
pp. 252-258 ◽  
Author(s):  
Robert Deville ◽  
Óscar Madiedo
Keyword(s):  
Nikodym Property

scholarly journals On the Radon-Nikodym Property for Vector Measures and Extensions of Transfunctions

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Piotr Mikusiński ◽  
John Paul Ward
Keyword(s):  
Banach Space ◽  
Bounded Variation ◽  
Vector Measure ◽  
Measurable Space ◽  
Unique Extension ◽  
Natural Conditions ◽  
Vector Measures ◽  
Nikodym Property

AbstractIf \left( {{\mu _n}} \right)_{n = 1}^\infty are positive measures on a measurable space (X, Σ) and \left( {{v_n}} \right)_{n = 1}^\infty are elements of a Banach space 𝔼 such that \sum\nolimits_{n = 1}^\infty {\left\| {{v_n}} \right\|{\mu _n}\left( X \right)} < \infty, then \omega \left( S \right) = \sum\nolimits_{n = 1}^\infty {{v_n}{\mu _n}\left( S \right)} defines a vector measure of bounded variation on (X, Σ). We show 𝔼 has the Radon-Nikodym property if and only if every 𝔼-valued measure of bounded variation on (X, Σ) is of this form. This characterization of the Radon-Nikodym property leads to a new proof of the Lewis-Stegall theorem.We also use this result to show that under natural conditions an operator defined on positive measures has a unique extension to an operator defined on 𝔼-valued measures for any Banach space 𝔼 that has the Radon-Nikodym property.

scholarly journals A Variational McShane Characterization of Locally Convex Spaces Possessing the Radon-Nikodym Property

2014 ◽  
Vol 58 (1) ◽  
pp. 23-35
Author(s):  
Sokol Bush Kaliaj
Keyword(s):  
Lebesgue Measure ◽  
Vector Spaces ◽  
Locally Convex Spaces ◽  
Absolutely Continuous ◽  
Topological Vector Spaces ◽  
Nikodym Property ◽  
Interval Functions ◽  
Locally Convex ◽  
Convex Spaces

Abstract We present a characterization of complete locally convex topological vector spaces possessing the Radon-Nikodym property in terms of additive interval functions whose McShane variational measures are absolutely continuous with respect to the Lebesgue measure.

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