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Lattice separation, coseparation and regular measures. (English) Zbl 0857.28011
Summary: Let $$X$$ be an arbitrary non-empty set, and let $$\mathcal L$$, $${\mathcal L}_1$$, $${\mathcal L}_2$$ be lattices of subsets of $$X$$ containing $$\emptyset$$ and $$X$$. $${\mathcal A}({\mathcal L})$$ denotes the algebra generated by $$\mathcal L$$ and $$M({\mathcal L})$$ the set of finite, non-trivial, non-negative finitely additive measures on $${\mathcal A}({\mathcal L})$$. $$I({\mathcal L})$$ denotes those elements of $$M({\mathcal L})$$ which assume only the values zero and one. In terms of a $$\mu\in M({\mathcal L})$$ or $$I({\mathcal L})$$, various outer measures are introduced. Their properties are investigated. The interplay of measurability, smoothness of $$\mu$$, regularity of $$\mu$$ and lattice topological properties on these outer measures is also investigated.
Finally, applications of these outer measures to separation type properties between pairs of lattices $${\mathcal L}_1$$, $${\mathcal L}_2$$, where $${\mathcal L}_1\subset{\mathcal L}_2$$, are developed. In terms of measures from $$I({\mathcal L})$$, necessary and sufficient conditions are established for $${\mathcal L}_1$$ to semi-separate $${\mathcal L}_2$$, for $${\mathcal L}_1$$ to separate $${\mathcal L}_2$$, and finally for $${\mathcal L}_1$$ to coseparate $${\mathcal L}_2$$.
##### MSC:
 28C15 Set functions and measures on topological spaces (regularity of measures, etc.) 28A12 Contents, measures, outer measures, capacities 28A60 Measures on Boolean rings, measure algebras 06B30 Topological lattices
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