International Journal of Mathematics and Mathematical Sciences
Volume 22 (1999), Issue 4, Pages 713-726
Smoothness conditions on measures using Wallman spaces
St. John's University, Department of Mathematics and Computer Science, 8000 Utopia Parkway, Jamaica 11439, NY, USA
Received 15 October 1997; Revised 13 April 1998
Copyright © 1999 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.
In this paper, denotes an arbitrary nonempty set, a lattice of subsets of with is the algebra generated by and is the set of nontrivial, finite, and finitely additive measures on , and is the set of elements of which are -regular. It is well known that any induces a finitely additive measure on an associated Wallman space. Whenever is countably additive.
We consider the general problem of given , how do properties of imply smoothness properties of ? For instance, what conditions on are necessary and sufficient for to be -smooth on , or strongly -smooth on , or countably additive? We consider in discussing these questions either of two associated Wallman spaces.