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Extensions of tight set functions with applications in topological measure theory


Author: Wolfgang Adamski
Journal: Trans. Amer. Math. Soc. 283 (1984), 353-368
MSC: Primary 28A10; Secondary 28A12
DOI: https://doi.org/10.1090/S0002-9947-1984-0735428-9
MathSciNet review: 735428
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Abstract: Let $ {\mathcal{K}_1},\,{\mathcal{K}_2}$ be lattices of subsets of a set $ X$ with $ {\mathcal{K}_1} \subset {\mathcal{K}_2}$. The main result of this paper states that every semifinite tight set function on $ {\mathcal{K}_1}$ can be extended to a semifinite tight set function on $ {\mathcal{K}_2}$. Furthermore, conditions assuring that such an extension is uniquely determined or $ \sigma $-smooth at $ \phi $ are given. Since a semifinite tight set function defined on a lattice $ \mathcal{K}$ [and being $ \sigma $-smooth at $ \phi $] can be identified with a semifinite $ \mathcal{K}$-regular content [measure] on the algebra generated by $ \mathcal{K}$, the general results are applied to various extension problems in abstract and topological measure theory.


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DOI: https://doi.org/10.1090/S0002-9947-1984-0735428-9
Article copyright: © Copyright 1984 American Mathematical Society

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