A characterization of the Radon-Nikodym property for vector valued\n measures

2017/01/17 by Piotr Mikusiński, Mikusinski, Piotr, John Paul Ward +1
Mathematics · #Advanced Banach Space Theory #Mathematical Analysis and Transform Methods #Approximation Theory and Sequence Spaces

paper · pdf · doi:10.48550/arxiv.1701.04837

Abstract

If \μ₁,\μ₂,\… are positive measures on a measurable space\n(X,\Σ) and v₁,v₂, \… are elements of a Banach space mathbb E\nsuch that \∑_n=1^\∞ \‖v_n\‖ \μ_n(X) < \∞, then \ω (S)=\n\∑_n=1^\∞ v_n \μ_n(S) defines a vector measure of bounded variation\non (X,\Σ). We show mathbb E has the Radon-Nikodym property if and\nonly if every mathbb E-valued measure of bounded variation on (X,\Σ)\nis of this form.\n As an application of this result we show that under natural conditions an\noperator defined on positive measures, has a unique extension to an operator\ndefined on mathbb E-valued measures for any Banach space mathbb E\nthat has the Radon-Nikodym property.\n

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