Outer measures, measurability, and lattice regular measures

Abstract

Let X be an arbitrary non-empty set, and ℒ a lattice of subsets of X such that ∅, X∈ℒ. 𝒜(ℒ) denotes the algebra generated by ℒ and I(ℒ) those zero-one valued, non-trivial, finitely additive measures on 𝒜(ℒ)Iσ(ℒ) denotes those elements of I(ℒ) that are σ-smooth on ℒ, and IR(ℒ) denotes those elements of I(ℒ) that are ℒ-regular while IRσ(ℒ)=IR(ℒ)∩Iσ(ℒ). In terms of those and other subsets of I(ℒ), various outer measures are introduced, and their properties are investigated. Also, the interplay between the measurable sets associated with these outer measures, regularity properties of the measures, smoothness properties of the measures, and lattice topological properties are thoroughly investigated- yielding new results for regularity or weak regularity of these measures, as well as domination on a lattice of a suitably given measure by a regular one Finally, elements of Iσ(ℒ) are fully characterized in terms of induced measures on a certain generalized Wallman space.

Keywords

• associated outer measures
• measurable sets
• weakly regular measures
• slightly regular measures
• almost normal lattices
• domination by regular measures.