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

Absolutely continuous operators on function spaces and vector measures

Open Access


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.


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 


  1. 1.
    Aliprantis C.D., Burkinshaw O.: Positive Operators. Academic Press, New York (1985)MATHGoogle Scholar
  2. 2.
    Aliprantis, C.D., Burkinshaw, O.: Locally Solid Riesz Spaces with Applications to Economics, 2nd edn. In: Math. Surveys and Monographs, no. 105 (2003)Google Scholar
  3. 3.
    Bourbaki N.: Elements of Mathematics, Topological Vector Spaces, chaps. 1–5. Springer Verlag, Berlin (1987)CrossRefGoogle Scholar
  4. 4.
    Cristescu R.: Topological Vector Spaces. Editura Academiei, Romania (1977)MATHGoogle Scholar
  5. 5.
    Diestel, J., Uhl, J.J.: Vector Measures, Amer. Math. Soc., Math. Surveys 15, Providence, RI (1977)Google Scholar
  6. 6.
    Dodds P.G.: o-weakly compact mappings of Riesz spaces. Trans. Amer. Math. Soc. 214, 389–402 (1975)MathSciNetMATHGoogle Scholar
  7. 7.
    Drewnowski L.: Decompositions of set functions. Studia Math. 48, 23–48 (1973)MathSciNetMATHGoogle Scholar
  8. 8.
    Graves W.H., Ruess W.: Compactness in spaces of vector-valued measures and a natural Mackey topology for spaces of bounded measurable functions. Contemp. Math. 2, 189–203 (1980)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Hoffman-Jorgensen J.: Vector measures. Math. Scand. 28, 5–32 (1971)MathSciNetGoogle Scholar
  10. 10.
    Kantorovich L.V., Akilov A.V.: Functional Analysis. Pergamon Press, Oxford-Elmsford, New York (1982)MATHGoogle Scholar
  11. 11.
    Namioka, I.: Partially ordered linear topological spaces. Mem. Amer. Math. Soc. 24 (1957)Google Scholar
  12. 12.
    Orlicz W.: Operations and linear functionals in spaces of φ-integrable functions, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 8, 563–565 (1960)MathSciNetMATHGoogle Scholar
  13. 13.
    Orlicz W., Wnuk W.: Absolutely continuous and modularly continuous operators defined on spaces of measurable functions. Ricerche di Matematica 15(2), 243–258 (1991)MathSciNetGoogle Scholar
  14. 14.
    Panchapagesan T.V.: Applications of a theorem of Grothendieck to vector measures. J. Math. Anal. Appl. 214, 89–101 (1997)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Schaefer H., Zhang X-D.: On the Vitali–Hahn–Saks theorem, Operator Theory: Advances and Applications, vol. 75, pp. 289–297. Birkhäuser, Basel (1995)Google Scholar
  16. 16.
    Zhang X-D.: On weak compactness in spaces of measures. J. Funct. Anal. 143, 1–9 (1997)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© The Author(s) 2012

Authors and Affiliations

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

Personalised recommendations