## Applications of outer measures to separation properties of lattices and regular or $$\sigma$$-smooth measures.(English)Zbl 0841.28005

Summary: Associated with a 0-1 measure $$\mu\in I({\mathcal L})$$, where $$\mathcal L$$ is a lattice of subsets of $$X$$, are outer measures $$\mu'$$ and $$\widetilde\mu$$; associated with a $$\sigma$$-smooth 0-1 measure $$\mu\in I_\sigma({\mathcal L})$$ is an outer measure $$\mu''$$ or with $$\mu\in I_\sigma({\mathcal L}')$$, $${\mathcal L}'$$ being the complementary lattice, another outer measure $$\overset\approx \mu$$. These outer measures and their associated measurable sets are used to establish separation properties on $$\mathcal L$$ and regularity and $$\sigma$$-smoothness of $$\mu$$. Separation properties between two lattices $${\mathcal L}_1$$ and $${\mathcal L}_2$$, $${\mathcal L}_1\subseteq {\mathcal L}_2$$, are similarly investigated. Notions of strongly $$\sigma$$-smooth and slightly regular measures are also used.

### MSC:

 28A12 Contents, measures, outer measures, capacities 28C15 Set functions and measures on topological spaces (regularity of measures, etc.)
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