Remarks on μ″-measurbale sets: regularity, σ-smootheness, and measurability

Abstract

Let X be an arbitrary nonempty set and ℒ a lattice of subsets of X such that ϕ,X∈ℒ. 𝒜(ℒ) is the algebra generated by ℒ and ℳ(ℒ) denotes those nonnegative, finite, finitely additive measures μ on 𝒜(ℒ). I(ℒ) denotes the subset of ℳ(ℒ) of nontrivial zero-one valued measures. Associated with μ∈I(ℒ) (or Iσ(ℒ)) are the outer measures μ′ and μ″ considered in detail. In addition, measurability conditions and regularity conditions are investigated and specific characteristics are given for 𝒮μ″, the set of μ″-measurable sets. Notions of strongly σ-smooth and vaguely regular measures are also discussed. Relationships between regularity, σ-smoothness and measurability are investigated for zero-one valued measures and certain results are extended to the case of a pair of lattices ℒ1,ℒ2 where ℒ1⊂ℒ2.

Keywords

• Outer measure
• cover regular outer measure
• strongly σ-smooth and vaguely regular measures.