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Remarks on μ ″ -measurbale sets: regularity, σ -smootheness, and measurability

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  • Carman Vlad

Abstract

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 I σ ( 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 𝒮 μ ″ , 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 ℒ 1 , ℒ 2 where ℒ 1 ⊂ ℒ 2 .

Suggested Citation

  • Carman Vlad, 1999. "Remarks on μ ″ -measurbale sets: regularity, σ -smootheness, and measurability," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 22, pages 1-10, January.
    • Handle: RePEc:hin:jijmms:872748
    DOI: 10.1155/S0161171299223915
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