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Components in vector lattices and extreme extensions of quasi-measures and measures

Published online by Cambridge University Press:  18 May 2009

Z. Lipecki
Z. Lipecki
Affiliation:
Institute of MathematicsPolish Academy of SciencesWroclaw BranchKopernika 1851–617 WroclawPoland
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We develop some ideas contained in the author's paper [8] which was, in turn, inspired by Bierlein and Stich [5]. The main body of the present paper is divided into three sections. Section 2 is concerned with some vector-lattice-theoretical results. They are then applied to extensions of quasi-measures and measures in Sections 3 and 4, respectively.

Let X be a vector lattice, let x ε X+ and let S be a non-empty set. Theorems 1 and 2 describe some properties of the convex set

(see Section 2 for the definition of the sum above). The extreme points of Dx,s are characterized in terms of the components of x. It is also shown that if X has the principal projection property and S is countable, then extr Dx,s is, in some sense, large in Dx,s. Furthermore, for finite S, each point in Dx,s is then a sσ-convex combination of extreme ones.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 35 , Issue 2 , May 1993 , pp. 153 - 162
Copyright
Copyright © Glasgow Mathematical Journal Trust 1993

References

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