Let X be an arbitrary non-empty set, and let ℒ, ℒ1, ℒ2
be lattices of subsets of X
containing
ϕ and X. 𝒜(ℒ) designates the algebra generated by ℒ and M(ℒ), these finite, non-trivial,
non-negative finitely additive measures on 𝒜(ℒ). I(ℒ) denotes those elements of M(ℒ) which assume
only the values zero and one. In terms of a μ∈M(ℒ) or I(ℒ), various outer measures are introduced.
Their properties are investigated. The interplay of
measurability, smoothness of μ, regularity of μ and
lattice topological properties on these outer measures is also investigated.Finally, applications of these outer measures to separation type
properties between pairs of
lattices ℒ1, ℒ2 where ℒ1⊂ℒ2 are developed. In
terms of measures from I(ℒ), necessary and sufficient
conditions are established for ℒ1 to semi-separate ℒ2, for ℒ1 to separate ℒ2, and finally for ℒ1 to coseparate ℒ2.
