International Journal of Mathematics and Mathematical Sciences
Volume 18 (1995), Issue 4, Pages 725-734
Outer measures associated with lattice measures and their application
St. John's University, Department of Mathematics & Computer Science, Jamaica 11439, NY, USA
Received 18 March 1994; Revised 7 July 1994
Copyright © 1995 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.
Consider a set and a lattice of subsets of such that , . denotes
those bounded finitely additive measures on which are studied, and denotes those elements
of which are valued. Associated with a or a (the elements of
which are -smooth on ) are outer measures and . In terms of these outer measures various
regularity properties of can be introduced, and the interplay between regularity, smoothness, and
measurability is investigated for both the valued case and the more general case. Certain results
for the special case carry over readily to the more general case or with at most a regularity assumption
on or , while others do not. Also, in the special case of valued measures more refined
notions of regularity can be introduced which have no immediate analogues in the general case.