Lattice-valued fuzzy measure and lattice-valued fuzzy integral

Abstract

In this paper, (1) the concepts of lattice-valued fuzzy measure (with no valuation property) and lower (resp. upper) lattice-valued fuzzy integral are proposed, which give the unified discription to the fuzzy measures and fuzzy integrals studied by Delgado and Moral, Qiao, Ralescu, Adams, Sugeno, Wang, and Zhang; (2) some asymptotic structural characteristics of lattice-valued fuzzy measures are introduced, and some relations between them are given; (3) some concepts of convergences for lattice-valued functions are defined, and Riesz' theorem, Egoroff's theorems and Lebesgue's theorem for lattice-valued measurable function are proved; (4) the monotone increasing (resp. decreasing) convergence theorem and almost (resp. pseudo almost) everywhere convergence theorrem for lower (resp. upper) lattice-valued fuzzy integral are shown under some weak conditions.