Abstract

Associated with a 0−1 measure μ∈I(ℒ) where ℒ is a lattice of subsets of X are outer measures μ′ and μ˜; associated with a σ-smooth 0−1 measure μ∈Iσ(ℒ) is an outer measure μ″ or with μ∈Iσ(ℒ′), ℒ′ being the complementary lattice, another outer measure μ˜˜. These outer measures and their associated measurable sets are used to establish separation properties on ℒ and regularity and σ-smoothness of μ. Separation properties between two lattices ℒ1 and ℒ2, ℒ1⫅ℒ2, are similarly investigated. Notions of strongly σ-smooth and slightly regular measures are also used.