Coincidences of the Concave Integral and the Pan-Integral

Ouyang, Yao; Li, Jun; Mesiar, Radko

doi:10.3390/sym9060090

Cited by 18 publications

(3 citation statements)

References 30 publications

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“…In further research, we will focus on the application of these methods. For instance, we will generalize convergence in measure theorems of nonlinear integrals [17,18] in the general framework concerning a pair of monotone measures, and discuss the linearity of pan-integrals and concave integrals [19,20], our results will be useful.…”

Section: Discussionmentioning

confidence: 99%

Absolute Continuity of Fuzzy Measures and Convergence of Sequence of Measurable Functions

2020

Mathematics

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In this note, the convergence of the sum of two convergent sequences of measurable functions is studied by means of two types of absolute continuity of fuzzy measures, i.e., strong absolute continuity of Type I, and Type VI. The discussions of convergence a.e. and convergence in measure are done in the general framework relating to a pair of monotone measures, and general results are shown. The previous related results are generalized.

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Section: Discussionmentioning

confidence: 99%

Absolute Continuity of Fuzzy Measures and Convergence of Sequence of Measurable Functions

2020

Mathematics

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“…When κ is subadditive, the concave integral cav •dκ is positive linear on F + (Ouyang et al [46]). For more basic properties of concave integrals, see [3,37,40,[47][48][49].…”

Section: Concave Integralsmentioning

confidence: 99%

Decomposition Integrals of Set-Valued Functions Based on Fuzzy Measures

2023

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The decomposition integrals of set-valued functions with regards to fuzzy measures are introduced in a natural way. These integrals are an extension of the decomposition integral for real-valued functions and include several types of set-valued integrals, such as the Aumann integral based on the classical Lebesgue integral, the set-valued Choquet, pan-, concave and Shilkret integrals of set-valued functions with regard to capacity, etc. Some basic properties are presented and the monotonicity of the integrals in the sense of different types of the preorder relations are shown. By means of the monotonicity, the Chebyshev inequalities of decomposition integrals for set-valued functions are established. As a special case, we show the linearity of concave integrals of set-valued functions in terms of the equivalence relation based on a kind of preorder. The coincidences among the set-valued Choquet, the set-valued pan-integral and the set-valued concave integral are presented.

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“…In the above discussions, we only discussed the Sugeno integral and the Choquet integral. There are two other important nonlinear integrals, the pan-integral [7,11], and the concave integral [12,13] (see also [26,27]), as follows:…”

Section: Pan-integral and Concave Integralmentioning

confidence: 99%

On Null-Continuity of Monotone Measures

2020

Mathematics

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The null-continuity of monotone measures is a weaker condition than continuity from below and possesses many special properties. This paper further studies this structure characteristic of monotone measures. Some basic properties of null-continuity are shown and the characteristic of null-continuity is described by using convergence of sequence of measurable functions. It is shown that the null-continuity is a necessary condition that the classical Riesz’s theorem remains valid for monotone measures. When considered measurable space ( X , A ) is S-compact, the null-continuity condition is also sufficient for Riesz’s theorem. By means of the equivalence of null-continuity and property (S) of monotone measures, a version of Egoroff’s theorem for monotone measures on S-compact spaces is also presented. We also study the Sugeno integral and the Choquet integral by using null-continuity and generalize some previous results. We show that the monotone measures defined by the Sugeno integral (or the Choquet integral) preserve structural characteristic of null-continuity of the original monotone measures.

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Coincidences of the Concave Integral and the Pan-Integral

Cited by 18 publications

References 30 publications

Absolute Continuity of Fuzzy Measures and Convergence of Sequence of Measurable Functions

Absolute Continuity of Fuzzy Measures and Convergence of Sequence of Measurable Functions

Decomposition Integrals of Set-Valued Functions Based on Fuzzy Measures

On Null-Continuity of Monotone Measures

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