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Proceedings of the American Mathematical Society

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Lattices with the Alexandrov properties

Author: Albert Gorelishvili
Journal: Proc. Amer. Math. Soc. 114 (1992), 1045-1049
MSC: Primary 28A33; Secondary 28A12, 28C15
MathSciNet review: 1094503
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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.

References [Enhancements On Off] (What's this?)

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Article copyright: © Copyright 1992 American Mathematical Society