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On the Vitali–Hahn–Saks theorem

Published online by Cambridge University Press:  14 November 2011

Aníbal Moltó
Aníbal Moltó
Affiliation:
Facultad de Matemáticas, Dr Moliner s/n, Burjasot, Valencia, Spain
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Synopsis

In this paper, a class of Boolean rings containing the class discussed in papers by Seever (1968) and Faires (1976), is defined in such a way that an extension of the classical Vitali–Hahn–Saks theorem holds for exhausting additive set functions. Some new compact topological spaces K for which C(K) is a Grothendieck space are constructed and a Nikodym type theorem is deduced from it. The Boolean algebras of Seever and Faires and those we study here are defined by ‘interpolation properties’ between disjoint sequences in the algebra. We give an example at the end of the paper that illustrates the difficulties arising when we try to find a larger class of Boolean algebras, defined in terms of such properties, for which the Vitali–Hanh–Saks theorem holds.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 90 , Issue 1-2 , 1981 , pp. 163 - 173
Copyright
Copyright © Royal Society of Edinburgh 1981

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