Asymptotic Structural Characteristics of Fuzzy Measure and their Applications

Abstract

In the fuzzy measure theory, as Sugeno’s fuzzy measures lose the additivity in general, the concept “almost”, which is well known in the classical measure theory, splits into two different concepts “almost” and “pseudo-almost”. It complicates the relations among the convergences of the sequence of measurable functions on the fuzzy measure space, and increases the content of the theory of convergences of the seauence of fuzzy integrals. In order to replace the additivity, it is quite necessary to investigate some asymptotic behaviors of a fuzzy measure at the sequences of sets which are called “waxing” and “waning”, and to introduce some new concepts, such as “autocontinuity”, “converse-autocontinuity” and “pseudo-autocontinuity”. Ihey describe some asymptotic structural characteristics of a fuzzy measure. Bу means of them, we give four forms of generalization for each of the Egoroff’s theorem, the Liesz’s theorem and the Lebesgne’s theorem respectively, and prove a lot of convergence theorems on the sequence of fuzzy integrals.