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

Abstract

Abstract If ( μ n ) n = 1 ∞ \left( {{\mu _n}} \right)_{n = 1}^\infty are positive measures on a measurable space (X, Σ) and ( v n ) n = 1 ∞ \left( {{v_n}} \right)_{n = 1}^\infty are elements of a Banach space 𝔼 such that ∑ n = 1 ∞ ‖ v n ‖ μ n ( X ) < ∞ \sum\nolimits_{n = 1}^\infty {\left\| {{v_n}} \right\|{\mu _n}\left( X \right)} < \infty , then ω ( S ) = ∑ n = 1 ∞ v n μ n ( S ) \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.