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