International Journal of Mathematics and Mathematical Sciences
Volume 19 (1996), Issue 2, Pages 253-262
doi:10.1155/S016117129600035X
Applications of outer measures to separation properties of lattices and regular or σ-smooth measures
University of Maine, Orono 04669-5752, Maine, USA
Received 1 March 1994; Revised 18 June 1994
Copyright © 1996 Pao-Sheng Hsu. 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
Associated with a 0−1 measure μ∈I(ℒ) where ℒ is a lattice of subsets of X are outer measures μ′ and μ˜; associated with a σ-smooth 0−1 measure μ∈Iσ(ℒ) is an outer measure μ″ or with μ∈Iσ(ℒ′), ℒ′ being the complementary lattice, another outer measure μ˜˜. These outer measures and their associated measurable sets are used to establish separation properties on ℒ and regularity and σ-smoothness of μ. Separation properties between two lattices ℒ1 and ℒ2, ℒ1⫅ℒ2, are similarly investigated. Notions of strongly σ-smooth and slightly regular measures are also used.