, Volume 17, Issue 3, pp 525-533,
Open Access This content is freely available online to anyone, anywhere at any time.

Absolutely continuous operators on function spaces and vector measures

Abstract

Let (Ω, Σ, μ) be a finite atomless measure space, and let E be an ideal of L 0(μ) such that ${L^\infty(\mu) \subset E \subset L^1(\mu)}$ . We study absolutely continuous linear operators from E to a locally convex Hausdorff space ${(X, \xi)}$ . Moreover, we examine the relationships between μ-absolutely continuous vector measures m : Σ → X and the corresponding integration operators T m : L (μ) → X. In particular, we characterize relatively compact sets ${\mathcal{M}}$ in ca μ (Σ, X) (= the space of all μ-absolutely continuous measures m : Σ → X) for the topology ${\mathcal{T}_s}$ of simple convergence in terms of the topological properties of the corresponding set ${\{T_m : m \in \mathcal{M}\}}$ of absolutely continuous operators. We derive a generalized Vitali–Hahn–Saks type theorem for absolutely continuous operators T : L (μ) → X.