The Properties And Structural Properties Of K-quasi-additive Fuzzy Integral

The concepts of quasi-additive and quasi-multiplication operators were proposed for the first time by Sugeno in 1987, and quasi-additive fuzzy measures and integrals theoretical framework were established. Based on it, the operations about extended addition and multiplication for given k-operator and t-operator were defined. Meanwhile, two kinds of integrals were submitted:tk-integral and kt-integral. The tk-integral is the generalization of S-type integral, kt-Integral is the extension of the Lebesgue integrals with respect to semi-measure, particularly gives transformation theorems of this type of integrals. Based on it, takes t=k, the induced operators were introduced in, the so-called k-quasi-additive fuzzy integrals were established. This paper also proves that this kind of fuzzy integrals is just a kind of k-quasi-additive fuzzy measures. That is to say, this kind of integral is a set function satisfying the requests of k-quasi-additive fuzzy measure. Its structural characteristics and convergence theorems were concerned.In this paper, firstly, based on the definition of the k-quasi-additive fuzzy integrals and transformation theorems of the integrals, we discuss the properties of k-quasi-additive fuzzy integrals. According to transformation theorems of the integrals, we give some new properties of k-quasi-additive fuzzy integral, so we perfect the properties of the integral. Secondly, based on the definition of the k-quasi-additive fuzzy integrals and transformation theorems. Study and discuss the structural characteristics such as monotony, weak-null- additivity, null-upper-continuity, continuous from above and continuous from below. Furthermore, we discuss the relationships between weak-null-additivity and null-additivity, and give proofs. More results are proved about the relationships between auto-continuity, uniformly auto-continuity, null-subtractivity, null-additivity, continuous from above and continuous from below. The relation graphs of structural characteristics are shown. Finally, this paper gives the definition and theorem of convergence in measure and uniform convergence about the k-quasi-additive fuzzy integrals, even gives the integrated theoretic proof.