L 1(μ) for vector-valued measureμ

L 1(μ) for vector-valued measureμ Let μ be a measure from a σ-algebra of subsets of a set T into a sequentially complete Hausdorff topological vector space X. Assume that μ is convexly bounded, i.e., the convex hull of its range is bounded in X, and denote by L 1(μ) the space of scalar valued functions on T which are integrable with respect to the vector measure μ. We study the inheritance of some properties from X to L 1(μ). We show that the bounded multiplier property passes from X to L 1(μ). Answering a 1972 problem of Erik Thomas, we show that for a rather large class of F-spaces X the non-containment of c 0 passes from X to L 1(μ). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

L 1(μ) for vector-valued measureμ

Positivity , Volume 14 (4) – Oct 29, 2010
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Publisher
SP Birkhäuser Verlag Basel
Copyright
Copyright © 2010 by Springer Basel AG
Subject
Mathematics; Econometrics; Calculus of Variations and Optimal Control; Optimization; Potential Theory; Operator Theory; Fourier Analysis
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-010-0090-8
Publisher site
See Article on Publisher Site

Abstract

Let μ be a measure from a σ-algebra of subsets of a set T into a sequentially complete Hausdorff topological vector space X. Assume that μ is convexly bounded, i.e., the convex hull of its range is bounded in X, and denote by L 1(μ) the space of scalar valued functions on T which are integrable with respect to the vector measure μ. We study the inheritance of some properties from X to L 1(μ). We show that the bounded multiplier property passes from X to L 1(μ). Answering a 1972 problem of Erik Thomas, we show that for a rather large class of F-spaces X the non-containment of c 0 passes from X to L 1(μ).

Journal

PositivitySpringer Journals

Published: Oct 29, 2010

References

  • Optimal domains for L 0-valued operators via stochastic measures
    Curbera, G.P.; Delgado, O.

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