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AN EXTENSION RESULT FOR CONTINUOUS VALUATIONS

Published online by Cambridge University Press:  01 April 2000

M. ALVAREZ-MANILLA ,
A. EDALAT  and
N. SAHEB-DJAHROMI
M. ALVAREZ-MANILLA
Affiliation:
Department of Computing, Imperial College of Science, Technology and Medicine, 180 Queen's Gate, London SW7 2BZ; m.alvarez-manilla@doc.ic.ac.uk, ae@doc.ic.ac.uk
A. EDALAT
Affiliation:
Department of Computing, Imperial College of Science, Technology and Medicine, 180 Queen's Gate, London SW7 2BZ; m.alvarez-manilla@doc.ic.ac.uk, ae@doc.ic.ac.uk
N. SAHEB-DJAHROMI
Affiliation:
LaBRI, Université Bordeaux-I, 351 cours de la Libération, 33405 Talence, France; Nasser.Saheb@labri.u-bordeaux.fr
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Abstract

It is shown, by a simple and direct proof, that if a bounded valuation on a monotone convergence space is the supremum of a directed family of simple valuations, then it has a unique extension to a Borel measure. In particular, this holds for any directed complete partial order with the Scott topology. It follows that every bounded and continuous valuation on a continuous directed complete partial order can be extended uniquely to a Borel measure. The last result also holds for σ-finite valuations, but fails for directed complete partial orders in general.

Type
Research Article
Information
Journal of the London Mathematical Society , Volume 61 , Issue 2 , April 2000 , pp. 629 - 640
Copyright
The London Mathematical Society 2000

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