Abstract
Let (Ω, Σ, λ) be a finite complete measure space, (E, ξ) be a sequentially complete locally convex Hausdorff space and E′ be its topological dual. Let caλ (Σ, E) stand for the space of all λ-absolutely continuous measures m: Σ → E. We show that a uniformly bounded subset M of caλ (Σ, E) is uniformly λ-absolutely continuous if and only if for every equicontinuous subset D of E′, there exists a submultiplicative Young function φ such that the set \(\left\{ {\frac{{d\left( {e'om} \right)}}{{d\lambda }}:m \in M,e' \in D} \right\}\) is relatively weakly compact in the Orlicz space Lφ(λ). As a consequence, we present a generalized Vitali–Hahn–Saks theorem on the setwise limit of a sequence of λ-absolutely continuous vector measures in terms of Orlicz spaces.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J. Alexopoulos, De la Vallée-Poussin’s theorem and weakly compact sets in Orlicz spaces, Quaest. Math., 17 (1994), 231–238.
C. D. Aliprantis and O. Burkinshaw, Positive Operators, Academic Press (New York, 1985).
D. Barcenas and C. Finol, On vector measures, uniform integrability and Orlicz spaces, in: Vector Measures, Integration and Related Topics, Oper. Theory Adv. Appl., 201, Birkhäuser Verlag (Basel, 2010), pp. 51–57.
J. Bourgain, Dunford–Pettis operator on L 1 and the Radon–Nikodym property, Israel J. Math., 37 (1980), 34–47.
J. B. Cooper, The strict topology and spaces with mixed topologies, Proc. Amer. Math. Soc., 30 (1971), 583–592.
J. B. Cooper, Saks Spaces and Applications to Functional Analysis, 2nd ed., Mathematical Studies, vol. 139, North-Holland (Amsterdam, 1978).
J. Diestel, Sequences and Series in Banach Spaces, Graduate Texts in Math., vol. 92, Springer-Verlag (Berlin–Heidelberg, 1984).
J. Diestel and J. J. Uhl, Vector Measures, Math. Surveys, vol. 15, Amer. Math. Soc. (Providence, RI, 1977).
L. Drewnowski, Topological rings of sets, continuous sets of functions, integration, II, Bull. Acad. Polon. Sci., Sér. Sci. Math. Astronom. Phys., 20 (1972), 277–286.
N. Dunford and J. Schwartz, Linear Operators. I: General Theory, Interscience Publ. Inc. (New York, 1967).
W. H. Graves and W. Ruess, Compactness in spaces of vector-valued measures and a natural Mackey topology for spaces of bounded measurable functions, Contemp. Math., 2 (1980), 189–203.
J. Hoffmann-Jørgensen, Vector measures, Math. Scand., 28 (1971), 5–32.
K. Musiał, Topics in the theory of Pettis integration, School on Measure Theory and Real Analysis (Grado, 1991), Rend. Istit. Mat. Univ. Trieste, 23, (1991), 177–262.
M. Nowak, A characterization of the Mackey topology τ (L ∞, L 1), Proc. Amer. Math. Soc., 108 (1990), 683–689.
M. Nowak, Dunford–Pettis operators on the space of Bochner integrable functions, Banach Center Publ., 95 (2011), 353–358.
M. M. Rao and Z. D. Ren, Theory of Orlicz Spaces, Marcel Dekker Inc. (New York,1991).
W. Ruess, [Weakly] compact operators and DF-spaces, Pacific J. Math., 98 (1982), 419–441.
J. J. Uhl, A characterization of strongly measurable Pettis integrable functions, Proc. Amer. Math. Soc., 34 (1972), 425–427.
A. Wiweger, Linear spaces with mixed topology, Studia Math., 20 (1961), 47–68.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Nowak, M. Applications of the Theory of Orlicz Spaces to Vector Measures. Anal Math 45, 111–120 (2019). https://doi.org/10.1007/s10476-018-0405-8
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10476-018-0405-8