# Absolutely continuous operators on function spaces and vector measures

Absolutely continuous operators on function spaces and vector measures 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# Absolutely continuous operators on function spaces and vector measures

, Volume 17 (3) – Jun 26, 2012
9 pages

/lp/springer_journal/absolutely-continuous-operators-on-function-spaces-and-vector-measures-34udjz8vJq
Publisher
Springer Journals
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-012-0187-3
Publisher site
See Article on Publisher Site

### 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.

### Journal

PositivitySpringer Journals

Published: Jun 26, 2012

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