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.

Keywords

• Outer measures associated with lattice measures
• lattice separation and coseparation
• weak regularity properties of measures.