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

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