International Journal of Mathematics and Mathematical Sciences
Volume 22 (1999), Issue 4, Pages 713-726
doi:10.1155/S0161171299227135

Smoothness conditions on measures using Wallman spaces

Abstract

In this paper, X denotes an arbitrary nonempty set, ℒ a lattice of subsets of X with ∅,X∈ℒ,A(ℒ) is the algebra generated by ℒ and M(ℒ) is the set of nontrivial, finite, and finitely additive measures on A(ℒ), and MR(ℒ) is the set of elements of M(ℒ) which are ℒ-regular. It is well known that any μ∈M(ℒ) induces a finitely additive measure μ¯ on an associated Wallman space. Whenever μ∈MR(ℒ),μ¯ is countably additive.We consider the general problem of given μ∈MR(ℒ), how do properties of μ¯ imply smoothness properties of μ? For instance, what conditions on μ¯ are necessary and sufficient for μ to be σ-smooth on ℒ, or strongly σ-smooth on ℒ, or countably additive? We consider in discussing these questions either of two associated Wallman spaces.