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

We propose the set-valued pan-integral of set-valued function based on number-valued fuzzy measure and present some basic properties. By means of a preorder on the class of all nonempty set of R1, we show the monotonicity of set-valued pan-integral in the sense of the preorder. We introduce an equivalence relation based on the preorder and demonstrate the linearity of set-valued pan-integral in the sense of the equivalence relation. The relationships of the set-valued pan-integral and the set-valued Choquet integral are discussed. Chebyshev's inequality of set-valued pan-integrals is shown. An open problem concerning the linearity of the set-valued pan-integral is raised.& COPY; 2023 Elsevier B.V. All rights reserved.