## Components in vector lattices and extreme extensions of quasi-measures and measures.(English)Zbl 0786.28002

This paper develops some ideas from the author’s recent paper [Arch. Math. 58, 288-293 (1992; Zbl 0756.28003)] and connects in an elegant way extensions of positive additive or countably additive set functions with properties of abstract vector lattices.
Let $$X$$ be a vector lattice. For $$x\in X_ +$$ and a nonempty set $$S$$ define ${\mathcal D}_{x,S}:=\left\{ (x_ s)_{s\in S}\in (X_ +)^ S\left| \sum_{s\in S} x_ s= x\right.\right\},$
${\mathcal C}_{x,S}:=\bigl\{(x_ s)_{s\in S}\in {\mathcal D}_{x,S}\mid x_ s\land x_ t=0\text{ whenever } s\neq t\bigr\}.$ Theorem 1 asserts that $${\mathcal C}_{x,S}$$ consists precisely of the extreme points of $${\mathcal D}_{x,S}$$, and Theorem 2 provides further information on $${\mathcal C}_{x,S}$$ in the case where $$X$$ has the principal projection property and $$S$$ is finite or countable.
These results apply to the order complete vector lattice $$ba({\mathcal M})$$, where $$\mathcal M$$ is an algebra of subsets of a set $$\Omega$$. For $$\mu\in ba({\mathcal M})_ +$$, the positive additive extensions of $$\mu$$ to the algebra generated by $$\mathcal M$$ and a finite partition $$(E_ s)_{s\in S}$$ of $$\Omega$$ are identified with an extreme subset of $${\mathcal D}_{\mu,S}$$ and this identification together with Theorems 1 and 2 yields a characterization and properties of the extreme extensions of $$\mu$$.
Corresponding results are obtained for countably additive set functions, countable partitions, and $$\sigma$$-algebras.

### MSC:

 28A12 Contents, measures, outer measures, capacities 06F20 Ordered abelian groups, Riesz groups, ordered linear spaces

Zbl 0756.28003
Full Text:

### References:

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