VITALI–HAHN–SAKS THEOREM FOR VECTOR MEASURES ON OPERATOR ALGEBRAS

We show that the direct generalization of the Vitali–Hahn–Saks theorem is not valid for all measures on von Neumann algebras. By applying a general equicontinuity argument, we prove a direct extension of the Vitali–Hahn–Saks theorem for a wide range of vector measures on von Neumann algebra s and JBW algebras. We also characterize relatively compact sets of vector measures on operator algebras.