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Consider a set X and a lattice ℒ of subsets of X such that ϕ, X∈ℒ. M(ℒ) denotesthose bounded finitely additive measures on A(ℒ) which are studied, and I(ℒ) denotes those elementsof 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 variousregularity properties of μ can be introduced, and the interplay between regularity, smoothness, andmeasurability is investigated for both the 0−1 valued case and the more general case. Certain resultsfor the special case carry over readily to the more general case or with at most a regularity assumptionon μ′ or μ″, while others do not. Also, in the special case of 0−1 valued measures more refinednotions of regularity can be introduced which have no immediate analogues in the general case

Outer measures associated with lattice measures and their application