Abstract
Given a pan-space and a nonnegative measurable function f on measurable space (X, A), the pan-integral of f with respect to monotone measure μ and pan-operation determines a new monotone measure on (X, A). Such the new monotone measure is absolutely continuous with respect to the monotone measure μ. We show that the new monotone measure preserves some important structural characteristics of the original monotone measures, such as continuity from below, subadditivity, null-additivity, weak null-additivity and (S) property. Since the pan-integral based on a pair of pan-operations covers the Sugeno integral (based on ) and the Shilkret integral (based on ), therefore, the previous related results for the Sugeno integral are covered by the results presented here, in the meantime, some special results related the Shilkret integral are also obtained.
Highlights
In generalized measure and integral theory, there are several kinds of important nonlinear integrals, the Choquet integral [1], the Sugeno integral [2], the pan-integral [3] and the concave integral introduced by Lehrer [4], etc
In general the monotone measures lose additivity, and they appear to be much looser than the classical measures
We have shown that the pan-integral with respect to the subadditive monotone measure is monotone superadditive functional
Summary
In generalized measure and integral theory, there are several kinds of important nonlinear integrals, the Choquet integral [1], the Sugeno integral [2], the pan-integral [3] and the concave integral introduced by Lehrer [4], etc. (see [5]). The pan-integral with respect to monotone measure μ relates to a commut-. ( ) ative isotonic semiring R+ , ⊕, ⊗ , where ⊕ is a pan-addition and ⊗ is a pan-multiplication related by the distributivity property (see [6]) This integral generalizes the Lebesgue integral and the Sugeno integral. ( ) dering a σ-additive measure m and the commutative isotonic semiring R+ , +,⋅ , the Lebesgue integral coincides with the pan-integral; when considering a mo-. ( ) notone measure μ and the commutative isotonic semiring R+ , ∨, ∧ , the Sugeno integral is recovered by the pan-integral with respect to (∨, ∧). In general the monotone measures lose additivity, and they appear to be much looser than the classical measures. Since the pan-integral covers the Sugeno integral, while the Shilkret integral, the previous related results for the Sugeno integral become special cases of the results presented here, in the meantime, some special results related the Shilkret integral are shown
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