Choquet Integral Defined By Lebesgue-Stieltjes Integral And Monotone Set-Valued Functions Defined By Set-Valued Integrals

This paper researches a new integral, that is, Choquet integral defined by Lebesgue-Stieltjes integral, and sevaral important structual properties of monotone set-valued function defined by set-valued integral about the original monotone set function are discussed. This paper is organized as follows:(1) Choquet integral draws much attention recently, many experts have investigated it, this paper generalizes Choquet integral by Lebesgue-Stieltjes measure, then Choquet integral defined by Lebesgue-Stieltjes integral is defined. Some properties of this new integral and convergence theorem, such as Monotone convergence theorem, Fatou lemma, Control convergence and so on, are discussed. It also researches the relationship between Choquet integral defined by Lebesgue-Stieltjes integral and Choquet integral, so integral conversion theorem is shown.(2) Similar to Aumann integral, it generalizes Lebesgue-Stieltjes integral defined by Choquet integral to set-valued case and discusses some properties and convergence theorem.(3) It gives some structual properties about set-valued function, such as null-additivity, strongly order continuity, property(S) and pseudometric generating property, then it discusses some properties of set-valued function defined by set-valued fuzzy integral and Choquet integral respectively, and it answers the open problem given by Guo and Zhang in《On set-valued fuzzy measure 》 ( Fuzzy Sets and Systems 160 (2004) 13-25).(4) Absolutely continuity plays an important role in measure theorem, this paper introduces several absolutely continuous, and shows that set-valued functiom defined by set-valued integral also has these continuity about original function.