Lattices with the Alexandrov properties

Abstract:

By an Alexandrov lattice we mean a $\delta$ normal lattice $\mathcal {L}$ of subsets of an abstract set $X$, such that the set of $\mathcal {L}$-regular countably additive bounded measures, denoted by $\operatorname {MR}(\sigma ,\mathcal {L})$, is sequentially closed in the set of $\mathcal {L}$-regular finitely additive bounded measures on the algebra generated by $\mathcal {L}$, i.e., if ${\mu _n} \in \operatorname {MR}(\sigma ,\mathcal {L})$ and ${\mu _n} \to \mu$ (weakly) then $\mu \in \operatorname {MR}(\sigma ,\mathcal {L})$. 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 $T:X \to Y,{\mathcal {L}_1}$ and ${\mathcal {L}_2}$ are lattices of subsets of $X$ of $Y$ respectively, and $T$ is ${\mathcal {L}_1} - {\mathcal {L}_2}$ continuous.