Remarks on Μ′′-measurable Sets: Regularity, Σ -smoothness, and Measurability

Let X be an arbitrary nonempty set and a lattice of subsets of X such that φ, X∈ . ( ) is the algebra generated by and ( ) denotes those nonnegative, finite, finitely additive measures μ on ( ). I( ) denotes the subset of ( ) of nontrivial zeroone valued measures. Associated with μ ∈ I( ) (or Iσ ( )) 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 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 1, 2 where 1 ⊂ 2.