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

Tong Kang, Leifan Yan, Long Ye, Jun Li

Abstract: 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 (+,⋅)<math><mo stretchy="false" is="true">(</mo><mo linebreak="badbreak" linebreakstyle="after" is="true">+</mo><mo is="true">,</mo><mo is="true">⋅</mo><mo stretchy="false" is="true">)</mo></math> 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 (⊕,⊗)<math><mo stretchy="false" is="true">(</mo><mo is="true">⊕</mo><mo is="true">,</mo><mo is="true">⊗</mo><mo stretchy="false" is="true">)</mo></math> is introduced in a natural way. It covers the previous set-valued pan-integral based on a fuzzy measure and the arithmetic operations pair (+,⋅)<math><mo stretchy="false" is="true">(</mo><mo linebreak="badbreak" linebreakstyle="after" is="true">+</mo><mo is="true">,</mo><mo is="true">⋅</mo><mo stretchy="false" is="true">)</mo></math>, and the set-valued Sugeno integral w.r.t. a fuzzy measure and the logical operations pair (∨,∧)<math><mo stretchy="false" is="true">(</mo><mo is="true">∨</mo><mo is="true">,</mo><mo is="true">∧</mo><mo stretchy="false" is="true">)</mo></math>.