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Compactness and Weak Compactness in Spaces of Compact-Range Vector Measures

Published online by Cambridge University Press:  20 November 2018

William H. Graves and
Wolfgang Ruess
William H. Graves
Affiliation:
University of North Carolina, Chapel Hill, North Carolina
Wolfgang Ruess
Affiliation:
Universität Essen, Essen, Federal Republic of Germany
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This paper features strong and weak compactness in spaces of vector measures with relatively compact ranges in Banach spaces. Its tools are the measure-operator identification of [16] and [24] and the description of strong and weak compactness in spaces of compact operators in [10], [11], and [29].

Given a Banach space X and an algebra of sets, it is shown in [16] that under the usual identification via integration of X-valued bounded additive measures on with X-valued sup norm continuous linear operators on the space of -simple scalar functions, the strongly bounded, countably additive measures correspond exactly to those operators which are continuous for the coarser (locally convex) universal measure topology τ on . It is through the latter identification that the results on strong and weak compactness in [10], [11], and [29] can be applied to X-valued continuous linear operators on the generalized DF space to yield results on strong and weak compactness in spaces of vector measures.

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 36 , Issue 6 , 01 December 1984, pp. 1000 - 1020
Copyright
Copyright © Canadian Mathematical Society 1984

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