On Constructing Finite, Finitely Subadditive Outer Measures, and Submodularity

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.