On the Radon-Nikodym property for vector measures and extensions of transfunctions

Piotr Mikusiński; John Paul Ward

Published: 2020-10-06

Vol. 35 No. 1 (2021)

On the Radon-Nikodym property for vector measures and extensions of transfunctions

Piotr Mikusiński

, John Paul Ward

Abstract

If (μ_n)_n=1^∞ are positive measures on a measurable space (X,Σ) and (v_n)_n=1^∞ are elements of a Banach space (X,Σ) such that Σ_n=1^∞‖v_n‖μ_n(X)<∞, then ω(S) = Σ_n=1^∞v_nμ_n(S) 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.

Keywords:

vector measures , Radon-Nikodym property , transfunctions

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Mikusiński, P., & Ward, J. P. (2020). On the Radon-Nikodym property for vector measures and extensions of transfunctions. Annales Mathematicae Silesianae, 35(1), 77–89. Retrieved from https://journals.us.edu.pl/index.php/AMSIL/article/view/13475

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