On Alexandrov lattices (Q1204371)

Summary: By an Alexandrov lattice we mean a \(\delta\)-normal lattice of subsets of an abstract set \(X\), such that the set of \({\mathcal L}\)-regular countably additive bounded measures is sequentially closed in the set of \({\mathcal L}\)-regular finitely additive bounded measures on the algebra generated by \(\mathcal L\) with the weak topology. For a pair of lattices \({\mathcal L}_ 1\subset{\mathcal L}_ 2\) in \(X\) sufficient conditions are indicated to determine when \({\mathcal L}_ 1\) Alexandrov implies that \({\mathcal L}_ 2\) is also Alexandrov and vice versa. The extension of this situation is given where \(T: X\to Y\) and \({\mathcal L}_ 1\) and \({\mathcal L}_ 2\) are lattices of subsets of \(X\) and \(Y\), respectively, and \(T\) is \({\mathcal L}_ 1-{\mathcal L}_ 2\) continuous.