International Journal of Mathematics and Mathematical Sciences
Volume 18 (1995), Issue 4, Pages 725-734
doi:10.1155/S0161171295000937

Outer measures associated with lattice measures and their application

Abstract

Consider a set X and a lattice ℒ of subsets of X such that ϕ, X∈ℒ. M(ℒ) denotes those bounded finitely additive measures on A(ℒ) which are studied, and I(ℒ) denotes those elements of M(ℒ) which are 0−1 valued. Associated with a μ∈M(ℒ) or a μ∈Mσ(ℒ) (the elements of M(ℒ) which are σ-smooth on ℒ) are outer measures μ′ and μ″. In terms of these outer measures various regularity properties of μ can be introduced, and the interplay between regularity, smoothness, and measurability is investigated for both the 0−1 valued case and the more general case. Certain results for the special case carry over readily to the more general case or with at most a regularity assumption on μ′ or μ″, while others do not. Also, in the special case of 0−1 valued measures more refined notions of regularity can be introduced which have no immediate analogues in the general case.