Lattices with the Alexandrov properties
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- by Albert Gorelishvili PDF
- Proc. Amer. Math. Soc. 114 (1992), 1045-1049 Request permission
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
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A. D. Alexandrov, Additive set-functions in abstract spaces, Mat. Sb. (N.S.) (a) 8(50) (1940), 307-348; (b) 9(51) (1941), 563-628; (c) 13 (55) (1943), 169-238.
- George Bachman and Panagiotis D. Stratigos, Some applications of the adjoint to lattice regular measures, Ann. Mat. Pura Appl. (4) 138 (1984), 379–397. MR 779552, DOI 10.1007/BF01762553
- George Bachman and Alan Sultan, On regular extensions of measures, Pacific J. Math. 86 (1980), no. 2, 389–395. MR 590550
- Harald Bergström, Weak convergence of measures, Probability and Mathematical Statistics, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1982. MR 690579
- Gustave Choquet, Theory of capacities, Ann. Inst. Fourier (Grenoble) 5 (1953/54), 131–295 (1955). MR 80760 V. S. Varadarajan, Measures on topological spaces, Amer. Math. Soc. Transi., vol. 48, Amer. Math. Soc., Providence, RI, 1965, pp. 161-228.
Additional Information
- © Copyright 1992 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 114 (1992), 1045-1049
- MSC: Primary 28A33; Secondary 28A12, 28C15
- DOI: https://doi.org/10.1090/S0002-9939-1992-1094503-2
- MathSciNet review: 1094503