Abstract
In this paper we generalize the classical Vitali-Hahn-Saks theorem to sets of countably additive vector measures which are compact in the strong operator topology. The main result asserts that a set of countably additive vector measures which is compact in the strong operator topology is uniformly countably additive. We accomplish this by first studying the properties of linear operators from Y *, the dual of a Banach space Y, into a Banach space X which are continuous with respect to the Mackey topology τ(Y *,Y) on Y* and the norm topology on X, and then applying the results to the special case where Y = L 1(μ) and Y * = L ∞(μ). Other related results on vector measures are also included.
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Dedicated to Professor A. C. Zaanen on the occasion of his eightieth birthday
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© 1995 Birkhäuser Verlag Basel/Switzerland
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Schaefer, H.H., Zhang, XD. (1995). On The Vitali-Hahn-Saks Theorem. In: Huijsmans, C.B., Kaashoek, M.A., Luxemburg, W.A.J., de Pagter, B. (eds) Operator Theory in Function Spaces and Banach Lattices. Operator Theory Advances and Applications, vol 75. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-9076-2_15
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DOI: https://doi.org/10.1007/978-3-0348-9076-2_15
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-0348-9896-6
Online ISBN: 978-3-0348-9076-2
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