Operator-valued measures and linear operators

We study operator-valued measures m:Σ →L(X,Y ), where L(X,Y ) stands for the space of all continuous linear operators between real Banach spaces X and Y and Σ is a σ-algebra of sets. We extend the Bartle–Dunford–Schwartz theorem and the Orlicz–Pettis theorem for vector measures to the case of operator-valued measures. We generalize the classical Vitali–Hahn–Saks theorem to sets of operator-valued measures which are compact in the strong operator topology. © 2007 Elsevier Inc. All rights reserved.