International Journal of Mathematics and Mathematical Sciences
Volume 15 (1992), Issue 4, Pages 681-695
doi:10.1155/S0161171292000905
Some topologies on the set of lattice regular measures
Long Island University, Brooklyn 11201, New York, USA
Received 20 December 1989; Revised 27 November 1990
Copyright © 1992 Panagiotis D. Stratigos. 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.
Abstract
We consider the general setting of A.D. Alexandroff, namely, an arbitrary set X and an arbitrary lattice of subsets of X, ℒ. 𝒜(ℒ) denotes the algebra of subsets of X generated by ℒ and MR(ℒ) the set of all lattice regular, (finitely additive) measures on 𝒜(ℒ).
First, we investigate various topologies on MR(ℒ) and on various important subsets of MR(ℒ), compare those topologies, and consider questions of measure repleteness whenever it is appropriate.
Then, we consider the weak topology on MR(ℒ), mainly when ℒ is δ and normal, which is the usual Alexandroff framework. This more general setting enables us to extend various results related to the special case of Tychonoff spaces, lattices of zero sets, and Baire measures, and to develop a systematic procedure for obtaining various topological measure theory results on specific subsets of MR(ℒ) in the weak topology with ℒ a particular topological lattice.