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In this paper, a new concept of fuzzy number-valued fuzzy integral of fuzzy number-valued function with respect to fuzzy number-valued fuzzy measure on the fuzzy set is introduced, and some equivalent definitions of the above-mentioned concept are given, and its elementary properties are discussed, and a series of the convergent theorems of fuzzy number-valued fuzzy integrals on the fuzzy set are obtained.

In this paper, some structural characteristics of the fuzzy number-valued fuzzy measure on the fuzzy σ-algebra are discussed. On the fuzzy number-valued fuzzy measure space, the concepts of ‘almost’ and ‘pseudo-almost’ are introduced, and Riesz's theorem, Lebesgue's theorem and Egoroff's theorem for sequences of fuzzy measurable functions and some convergence theorems for sequences of fuzzy number-valued fuzzy integrals on fuzzy sets are proved by some structural characteristics of the fuzzy number-valued fuzzy measure.

In this paper, some new concepts — the fuzzy number-valued fuzzy measure, autocontinuity and null-additivity of fuzzy number-valued fuzzy measures on the fuzzy σ-algebra, and the fuzzy number-valued fuzzy integral on the fuzzy set — are introduced. Some properties of fuzzy number-valued fuzzy integrals and some convergence theorems for sequences of fuzzy number-valued fuzzy integrals on a fuzzy set are discussed.

We introduce three important concepts of fuzzy complex numbers, fuzzy distance and fuzzy limit of fuzzy complex numbers, and give some elementary properties of the fuzzy complex numbers and fuzzy distance and fuzzy limit of the fuzzy complex numbers. We also discuss some important theorems of fuzzy complex numbers: nested closed rectangles theorem, Cauchy's criterion for convergence, accumulation principle, etc.

In this paper, we discuss some properties of a fuzzy number-valued fuzzy set function defined by a fuzzy number-valued fuzzy integral on the fuzzy set, and give sufficient conditions that it is a fuzzy number-valued fuzzy measure.

The concepts of possibility measure and consonant belief function on the fuzzy set are introduced and results analogous to those of Wang Zhenyuan (1985) are obtained.

In this paper, we introduce the concepts of fuzzy distance of fuzzy numbers, fuzzy limit of fuzzy numbers, fuzzy function and fuzzy limit of a fuzzy function and fuzzy continuous function on the set of fuzzy numbers and give some of their elementary properties. We also discuss some important theorems of fuzzy numbers and the fuzzy continuous function on the M -closed interval.