International Journal of Mathematics and Mathematical Sciences
Volume 16 (1993), Issue 1, Pages 49-59
doi:10.1155/S0161171293000055

On Alexandrov lattices

Abstract

By an Alexandrov lattice we mean a δ normal lattice of subsets of an abstract set X, such that the set of ℒ-regular countably additive bounded measures is sequentially closed in the set of ℒ-regular finitely additive bounded measures on the algebra generated by ℒ with the weak topology.For a pair of lattices ℒ1⊂ℒ2 in X sufficient conditions are indicated to determine when ℒ1 Alexandrov implies that ℒ2 is also Alexandrov and vice versa. The extension of this situation is given where T:X→Y and ℒ1 and ℒ2 are lattices of subsets of X and Y respectively and T is ℒ1−ℒ2 continuous.