On finitely subadditive outer measures and modularity properties

Abstract

Let ν be a finite, finitely subadditive outer measure on P(X). Define ρ (E)=ν (X)−ν (E′) for E⊂X. The measurable sets Sν and Sρ and the set S={E⊂X/ν (E)=ρ (E)} are investigated in general, and in the presence of regularity or modularity assumptions on ν. This is also done for ν0(E)=inf{ν (M)/E⊂M∈Sν }. General properties of ν are derived when ν is weakly submodular. Applications and numerous examples are given.