International Journal of Mathematics and Mathematical Sciences
Volume 22 (1999), Issue 4, Pages 713-726

Smoothness conditions on measures using Wallman spaces

Charles Traina

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, X denotes an arbitrary nonempty set, a lattice of subsets of X with ,X,A() is the algebra generated by and M() is the set of nontrivial, finite, and finitely additive measures on A(), and MR() is the set of elements of M() which are -regular. It is well known that any μM() induces a finitely additive measure μ¯ on an associated Wallman space. Whenever μMR(),μ¯ is countably additive.

We consider the general problem of given μMR(), 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 , or strongly σ-smooth on , or countably additive? We consider in discussing these questions either of two associated Wallman spaces.