Outer measures, measurability, and lattice regular measures
Let X be an arbitrary non-empty set, and ℒ a lattice of subsets of X such that ∅, X∈ℒ. 𝒜(ℒ) denotes the algebra generated by ℒ and I(ℒ) those zero-one valued, non-trivial, finitely additive measures on 𝒜(ℒ)Iσ(ℒ) denotes those elements of I(ℒ) that are σ-smooth on ℒ, and IR(ℒ) denotes those elements...
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| Main Author: | |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
1996-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Subjects: | |
| Online Access: | http://dx.doi.org/10.1155/S0161171296000488 |
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| Summary: | Let X be an arbitrary non-empty set, and ℒ a lattice of subsets of X such that
∅, X∈ℒ. 𝒜(ℒ) denotes the algebra generated by ℒ and I(ℒ) those zero-one valued, non-trivial, finitely
additive measures on 𝒜(ℒ)Iσ(ℒ) denotes those elements of I(ℒ) that are σ-smooth on ℒ, and IR(ℒ)
denotes those elements of I(ℒ) that are ℒ-regular while IRσ(ℒ)=IR(ℒ)∩Iσ(ℒ). In terms of those and
other subsets of I(ℒ), various outer measures are introduced, and their properties are investigated. Also,
the interplay between the measurable sets associated with these outer measures, regularity properties of
the measures, smoothness properties of the measures, and lattice topological properties are thoroughly
investigated- yielding new results for regularity or weak regularity of these measures, as well as
domination on a lattice of a suitably given measure by a regular one Finally, elements of Iσ(ℒ) are fully
characterized in terms of induced measures on a certain generalized Wallman space. |
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| ISSN: | 0161-1712 1687-0425 |