On Constructing Finite, Finitely Subadditive Outer Measures, and Submodularity - Charles Traina (International Journal of Mathematics and Mathematical Sciences)
Given a nonempty abstract set 𝑋, and a covering class 𝒞, and a finite, finitely subadditive outer measure 𝜈, we construct an outer measure 𝜈 and investigate conditions for 𝜈 to be submodular. We then consider several other set functions associated with 𝜈 and obtain conditions for equality of these functions on the lattice generated by 𝒞. Lastly, we describe a construction of a finite, finitely subadditive outer measure given an arbitrary family of subsets, ℬ, of 𝑋 and a nonnegative, finite set function 𝜏 defined on ℬ.