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International Journal of Mathematics and Mathematical Sciences
Volume 2008, Article ID 896480, 13 pages
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.


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, , of 𝑋 and a nonnegative, finite set function 𝜏 defined on .