International Journal of Mathematics and Mathematical Sciences
Volume 2008 (2008), Article ID 896480, 13 pages
doi:10.1155/2008/896480

On constructing finite, finitely subadditive outer measures, and submodularity

Abstract

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.