International Journal of Mathematics and Mathematical Sciences
Volume 16 (1993), Issue 1, Pages 33-40

On regular and sigma-smooth two valued measures and lattice generated topologies

Robert W. Shutz

P.O. Box 1149, West Babylon, N.Y. 11704, USA

Received 21 November 1991; Revised 7 April 1992

Copyright © 1993 Robert W. Shutz. 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.


Let X be an abstract set and L a lattice of subsets of X. I(L) denotes the non-trivial zero one valued finitely additive measures on A(L), the algebra generated by L, and IR(L) those elements of I(L) that are L-regular. It is known that I(L)=IR(L) if and only if L is an algebra. We first give several new proofs of this fact and a number of characterizations of this in topologicial terms.

Next we consider, I(σ*,L) the elements of I(L) that are σ-smooth on L, and IR(σ,L) those elements of I(σ*,L) that are L-regular. We then obtain necessary and sufficent conditions for I(σ*,L)=IR(σ,L), and in particuliar ,we obtain conditions in terms of topologicial demands on associated Wallman spaces of the lattice.