Lattice separation, coseparation and regular measures

Maurice C. Figueres

International Journal of Mathematics and Mathematical Sciences, 1996, vol. 19, 1-7

Abstract:

Let X be an arbitrary non-empty set, and let ℒ , ℒ 1 , ℒ 2 be lattices of subsets of X containing ϕ and X . 𝒜 ( ℒ ) designates the algebra generated by ℒ and M ( ℒ ) , these finite, non-trivial, non-negative finitely additive measures on 𝒜 ( ℒ ) . I ( ℒ ) denotes those elements of M ( ℒ ) which assume only the values zero and one. In terms of a μ ∈ M ( ℒ ) or I ( ℒ ) , various outer measures are introduced. Their properties are investigated. The interplay of measurability, smoothness of μ , regularity of μ 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 ℒ 1 , ℒ 2 where ℒ 1 ⊂ ℒ 2 are developed. In terms of measures from I ( ℒ ) , necessary and sufficient conditions are established for ℒ 1 to semi-separate ℒ 2 , for ℒ 1 to separate ℒ 2 , and finally for ℒ 1 to coseparate ℒ 2 .

Date: 1996
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:184758

DOI: 10.1155/S016117129600107X

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