Pan-integrals of set-valued functions based on fuzzy measures

Kang, Tong; Wu, Dongsheng; Li, Jun

doi:10.1016/j.fss.2023.108632

Cited by 2 publications

(4 citation statements)

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“…As we have seen, the set-valued decomposition integral is an extension of the decomposition integral for real-valued functions, and it unifies the previous set-valued integral schemes, including the Aumann integral [14], the set-valued Choquet integral ( [19,21]), the set-valued pan-integral ( [33]) and the set-valued concave integral, etc.…”

Section: Discussionmentioning

confidence: 85%

“…From Propositions 11 and 12 and similar to the proof of Proposition 4.2 in [33], it is not difficult to obtain the corresponding result for set-valued concave integrals as follows:…”

Section: Concave Integralsmentioning

confidence: 97%

“…When we take H = H Ch and H pan , respectively, then the set-valued decomposition integral H •dκ : R[Ω] → P(R 1 + ) goes back to the set-valued Choquet integral [19,21,32] and the set-valued pan-integral [33], respectively. In the following section, we will discuss the set-valued concave integral.…”

Section: Definition Of Set-valued Decomposition Integralsmentioning

confidence: 99%

“…More studies on the topics of set-valued functions and integrals of set-valued functions have also appeared in [23,[25][26][27][28][29][30]. In particular, the Choquet integrals of set-valued functions have been deeply investigated (see [19][20][21]31,32]), and the pan-integrals of set-valued functions have just been introduced and discussed in [33]. Noting that the decomposition integral forms a unified framework for the Choquet integrals, pan-integrals, concave integrals, etc., we naturally want to define a set-valued decomposition integral so that the previous set-valued integrals, such as the set-valued Choquet, the set-valued pan-and the set-valued concave integrals, become some special cases of set-valued decomposition integrals.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Discussionmentioning

confidence: 85%

“…From Propositions 11 and 12 and similar to the proof of Proposition 4.2 in [33], it is not difficult to obtain the corresponding result for set-valued concave integrals as follows:…”

Section: Concave Integralsmentioning

confidence: 97%