International Journal of Mathematics and Mathematical Sciences
Volume 19 (1996), Issue 4, Pages 773-779
doi:10.1155/S016117129600107X

Lattice separation, coseparation and regular measures

Abstract

Let X be an arbitrary non-empty set, and let ℒ, ℒ1, ℒ2 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 ℒ1, ℒ2 where ℒ1⊂ℒ2 are developed. In terms of measures from I(ℒ), necessary and sufficient conditions are established for ℒ1 to semi-separate ℒ2, for ℒ1 to separate ℒ2, and finally for ℒ1 to coseparate ℒ2.