Some notes on the pan-integrals of set-valued functions

This note answers an open problem which was proposed in the paper Kang et al. (2023) [9]. By means of subadditivity of fuzzy measure we show that the set-valued pan-integral based on a fuzzy measure and the operations pair (+, center dot) possesses the usual form of linearity. Thus, this open problem has a positive solution, and the previous result concerning the pseudo-linearity of set-valued pan-integrals is improved. We provide an example to show that the condition of subadditivity of fuzzy measure is necessary to the linearity of set-valued pan-integrals. The pan-integral of set-valued functions w.r.t. a fuzzy measure and pan-operations pair (circle plus, circle times) is introduced in a natural way. It covers the previous set-valued pan-integral based on a fuzzy measure and the arithmetic operations pair (+, center dot), and the set-valued Sugeno integral w.r.t. a fuzzy measure and the logical operations pair (proves, perpendicular to).

Automated summary

This note solves an open problem proposed in the paper Kang et al. (2023) [9] by demonstrating the linearity of set-valued pan-integrals based on a fuzzy measure and the operations pair (+, center dot) through the subadditivity of the fuzzy measure. It also provides an example to show the necessity of the subadditivity condition for the linearity of set-valued pan-integrals. Furthermore, it introduces the pan-integral of set-valued functions based on a fuzzy measure and pan-operations pair (circle plus, circle times).