International Journal of Mathematics and Mathematical Sciences
Volume 20 (1997), Issue 2, Pages 279-291
doi:10.1155/S0161171297000380
Outer measure analysis of topological lattice properties
Polytechnic University, Brooklyn 11201, New York, USA
Received 21 October 1994; Revised 8 September 1995
Copyright © 1997 Setiawati Wibisono. 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 X be a set and ℒ a lattice of subsets of X such that ∅, X∈ℒ. A(ℒ) is the algebra
generated by ℒ, M(ℒ) the set of nontrivial, finite, normegative, finitely additive measures on A(ℒ) and
I(ℒ) those elements of M(ℒ) which just assume the values zero and one. Various subsets of M(ℒ) and
I(ℒ) are included which display smoothness and regularity properties.
We consider several outer measures associated with dements of M(ℒ) and relate their behavior
to smoothness and regularity conditions as well as to various lattice topological properties. In addition,
their measurable sets are fully investigated. In the case of two lattices ℒ1, ℒ2, with ℒ1⊂ℒ2, we present
consequences of separation properties between the pair of lattices in terms of these outer measures, and
further demonstrate the extension of smoothness conditions on ℒ1 to ℒ2.