International Journal of Mathematics and Mathematical Sciences
Volume 19 (1996), Issue 2, Pages 343-350
doi:10.1155/S0161171296000488

Outer measures, measurability, and lattice regular measures

J. Ponnley1,2

1Department of Mathematics, Medgar Evers College, The City University of New York, 1650 Bedford Avenue, Brooklyn 11245-2298, New York, USA
2Department of Mathematical Sciences, Clark Atlanta University, Atlanta 30314, GA, USA

Received 28 September 1994; Revised 6 February 1995

Copyright © 1996 J. Ponnley. 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 an arbitrary non-empty set, and ℒ a lattice of subsets of X such that ∅, X∈ℒ. 𝒜(ℒ) denotes the algebra generated by ℒ and I(ℒ) those zero-one valued, non-trivial, finitely additive measures on 𝒜(ℒ)Iσ(ℒ) denotes those elements of I(ℒ) that are σ-smooth on ℒ, and IR(ℒ) denotes those elements of I(ℒ) that are ℒ-regular while IRσ(ℒ)=IR(ℒ)∩Iσ(ℒ). In terms of those and other subsets of I(ℒ), various outer measures are introduced, and their properties are investigated. Also, the interplay between the measurable sets associated with these outer measures, regularity properties of the measures, smoothness properties of the measures, and lattice topological properties are thoroughly investigated- yielding new results for regularity or weak regularity of these measures, as well as domination on a lattice of a suitably given measure by a regular one Finally, elements of Iσ(ℒ) are fully characterized in terms of induced measures on a certain generalized Wallman space.