Remarks on μ″-measurbale sets: regularity, σ-smootheness, and measurability
Let X be an arbitrary nonempty set and ℒ a lattice of subsets of X such that ϕ,X∈ℒ. 𝒜(ℒ) is the algebra generated by ℒ and ℳ(ℒ) denotes those nonnegative, finite, finitely additive measures μ on 𝒜(ℒ). I(ℒ) denotes the subset of ℳ(ℒ) of nontrivial zero-one valued measures. Associated with μ∈I(ℒ) (or...
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| Main Author: | |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
1999-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Subjects: | |
| Online Access: | http://dx.doi.org/10.1155/S0161171299223915 |
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| Summary: | Let X be an arbitrary nonempty set and ℒ a lattice of subsets of X such that ϕ,X∈ℒ. 𝒜(ℒ) is the algebra generated by ℒ and ℳ(ℒ) denotes those nonnegative, finite, finitely additive measures μ on 𝒜(ℒ). I(ℒ) denotes the subset of ℳ(ℒ) of nontrivial zero-one valued measures. Associated with μ∈I(ℒ) (or Iσ(ℒ)) are the outer measures μ′ and μ″ considered in detail. In addition, measurability conditions and regularity conditions are investigated and specific characteristics are given for 𝒮μ″, the set of μ″-measurable sets. Notions of strongly σ-smooth and vaguely regular measures are also discussed. Relationships between regularity, σ-smoothness and measurability are investigated for zero-one valued measures and certain results are extended to the case of a pair of lattices ℒ1,ℒ2 where ℒ1⊂ℒ2. |
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| ISSN: | 0161-1712 1687-0425 |