International Journal of Mathematics and Mathematical Sciences
Volume 2008 (2008), Article ID 896480, 13 pages
doi:10.1155/2008/896480
Research Article
On Constructing Finite, Finitely Subadditive Outer Measures, and Submodularity
Department of Mathematics & Computer Science, St. John's University, 8000 Utopia Parkway Queens, New York, NY 11439, USA
Received 1 August 2008; Accepted 3 December 2008
Academic Editor: Andrei Volodin
Copyright © 2008 Charles Traina. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.