Choquet integrals of set-valued functions with respect to set-valued fuzzy measures

Zhang, Deli; Guo, Chonghui

doi:10.1016/j.fss.2022.08.025

Cited by 6 publications

(2 citation statements)

References 39 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…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%

See 1 more Smart Citation

Decomposition Integrals of Set-Valued Functions Based on Fuzzy Measures

2023

View full text Add to dashboard Cite

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.

show abstract