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

If (μn)∞n=1 are positive measures on a measurable space (X, Σ) and (vn)∞n=1 are elements of a Banach space E such that Σ∞n=1 ||vn||μn(X) < ∞, then ω(S) = Σ∞n=1 vnμn(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.