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.
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Nowak, M. Absolutely continuous operators on function spaces and vector measures. Positivity 17, 525–533 (2013). https://doi.org/10.1007/s11117-012-0187-3
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DOI: https://doi.org/10.1007/s11117-012-0187-3
Keywords
- Function spaces
- Absolutely continuous operators
- Integration operators
- Countably additive vector measures
- Absolutely continuous vector measures
- Mackey topologies
- Order-bounded topology