Convergences And Choquet Integral Of Set-valued Functions On Fuzzy Measure Space

In this paper, some structural characteristics of nonadditive measure are studied. The convergence of the measurable functions (single-valued and set-valued) and some propertis of Choquet integral are discussed. It is organized as follows:(1) four kinds of continuity of monotone set function are introduced and four forms of generalization on monotone measure space for Lebesgue theorem are presented. The equivalence among the continuity from below and above of monotone set function and the monotone convergence theorems of fuzzy and of Choquet integrals are discussed, respectively. Dominated convergence theorem of Choquet integral of single-valued function is shown.(2) the convergence of the measurable closed-valued function (also called random set) is studied. Two forms of Egoroff theorem of the measurable closed-valued function on monotone measure space and on finite fuzzy measure space are proved, respectively.(3) the concept of Choquet integral of the measurable set-valued functions on fuzzy measure space is introduced . It is as a generalization of the Choquet integral of measurable single-valued functions The properties of Choquet integral are discussed. Monotone convergence theorem, Fatou's lemma and Lebesgue convergence theorem of Choquet integral are proved, respectively.