Mcshane Integral Of Both-branch-fuzzy-valued Functions And Its Generalization

The paper mainly has two parts.The first part:at first, based on Mcshane integral of interval-valued func-tions, Mcshane integral of both-branch-interval-valued functions is introduced. Then Mcshane integral of fuzzy-valued functions is extended to both-branch-fuzzy-valued functions and some of its basic properties are studied. At last, com-bining the definitions of both-branch-interval-valued functions and both-branch-fuzzy-valued functions, monotone convergence theorem and dominated conver-gence theorem for Mcshane integral of both-branch-fuzzy-valued functions are discussed.The second part:using the definition of traditional fine division on infinite interval and combining the relationship between fuzzy-valued functions on in-finite interval and its cut functions, Mcshane integral of fuzzy-valued functions on infinite interval is introduced. Furthermore, aiming at fuzzy-valued functions, the definition of equivalent measure fuzzy Mcshane integral and a condition of equivalence of its integrability are given. Last but not least, strong fuzzy Mcshane integral is defined and in the sense of this integral a necessary and sufficient condi-tion for its fuzzy valued function to be integrable is obtained. Thereby the theory of fuzzy integral is improved and enriched.