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Exhaustive operators and vector measures

Published online by Cambridge University Press:  20 January 2009

N. J. Kalton
N. J. Kalton
Affiliation:
University College of Swansea, Singleton Park, Swansea SA2 8PP
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Let S be a compact Hausdorff space and let Φ: C(S)→E be a linear operator defined on the space of real-valued continuous functions on S and taking values in a (real) topological vector space E. Then Φ is called exhaustive (7) if given any sequence of functions fn ∈ C(S) such that fn ≧ 0 and

then Φ(fn)→0 If E is complete then it was shown in (7) that exhaustive maps are precisely those which possess regular integral extensions to the space of bounded Borel functions on S; this is equivalent to possessing a representation

where μ is a regular countably additive E-valued measure defined on the σ-algebra of Borel subsets of S.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 19 , Issue 3 , March 1975 , pp. 291 - 300
Copyright
Copyright © Edinburgh Mathematical Society 1975

References

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