Positivity

, Volume 17, Issue 3, pp 525–533 | Cite as

Absolutely continuous operators on function spaces and vector measures

Open Access
Article

Abstract

Let (Ω, Σ, μ) be a finite atomless measure space, and let E be an ideal of L0(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 Tm : L(μ) → X. In particular, we characterize relatively compact sets \({\mathcal{M}}\) in caμ(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.

Keywords

Function spaces Absolutely continuous operators Integration operators Countably additive vector measures Absolutely continuous vector measures Mackey topologies Order-bounded topology 

Mathematics Subject Classification (2010)

46G10 28A33 28A25 47B38 

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Copyright information

© The Author(s) 2012

Authors and Affiliations

  1. 1.Faculty of Mathematics, Computer Science and EconometricsUniversity of Zielona GóraZielona GoraPoland

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