Extensions of tight set functions with applications in topological measure theory

Adamski, Wolfgang

doi:10.1090/S0002-9947-1984-0735428-9

Extensions of tight set functions with applications in topological measure theory
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Trans. Amer. Math. Soc. 283 (1984), 353-368 Request permission

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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