Continuous-from-above possibility measures and f-additive fuzzy measures on separable metric spaces: Characterization and regularity

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Abstract This paper characterizes continuous-from-above possibility measures and f-additive fuzzy measures on separable metric spaces. A set function μ is a continuous-from-above possibility measure on a non-empty set X [or an f-additive fuzzy measure on a separable metric space X ], if and only if there is a sequence {( x n , s n )} ⊃ X × [0,1] such that lim n →∞ s n = 0 and μ ( A ) = sup{ s n | x n ⊃ A } for every measurable set A . In addition, a continuous-from-above possibility measure on a T 1 -space and an f-additive fuzzy measure on a separable metric space are both regular, i.e., for any measurable set A and any ϵ > 0, there are open set G and closed set F such that F ⊃ A ⊃ G and μ ( A ) − ϵ ( F ) ⩽ μ ( F ) ⩽ μ ( G ) ⩽ μ ( A ) + ϵ .

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