International Journal of Mathematics and Mathematical Sciences
Volume 2003 (2003), Issue 8, Pages 461-474
doi:10.1155/S0161171203208115
On finitely subadditive outer measures and modularity properties
Department of Mathematics and Computer Science, St. John's University, 8000 Utopia Parkway, Jamaica 11439, NY, USA
Received 5 August 2002
Copyright © 2003 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
Let ν be a finite, finitely subadditive outer measure on
P(X). Define
ρ (E)=ν (X)−ν (E′) for E⊂X. The measurable sets
Sν and Sρ and the set S={E⊂X/ν (E)=ρ (E)} are investigated in general, and in the
presence of regularity or modularity assumptions on ν. This is also done for
ν0(E)=inf{ν (M)/E⊂M∈Sν }. General properties of ν are derived when ν is
weakly submodular. Applications and numerous examples are given.