T-fuzzy Valued Integral And A Note On K-quasi-additive Fuzzy Valued Integrals

The paper has two parts.The first part is about T-fuzzy valued integral.Firstly,on the basis of the generalized fuzzy integral,which is defined by nonnegative measurable functions. Through introducing T-norm operator,we extend the integrand functions to the fuzzy valued function and give the definition of so-called T-fuzzy valued integral. Then we achieve some basic properties of this integral. secondly,at satisfying certain conditions, we obtain some convergence theorem,which is similar with the classical Lebesgue integral. At last,we discuss T-fuzzy valued integral equation of this integral and gain some important results.The second part is about A note on K-quasi-additive fuzzy valued integral. Aiming at having established K-quasi-additive fuzzy valued integrals, we regard all of this kind of integrals as set functions being taken valued on measurable spaces. Applying the integrals transformation theorem and properties of inductive operators,we research the double-null asymptotic additivity and exhaustivity of this integral. These properties have very important significance for describing the convergence of sequence of fuzzy valued measurable functions and sequence of fuzzy integral.