Convergence Theorems On (G) Fuzzy Integrals

In this dissertation, We mainly discuss the convergence theory of generalized fuzzy integrals and some properties of sequences of generalized fuzzy integrable functions. The paper is organized as follows.(1) In chapter 1, We recommend some concepts on fuzzy measures and fuzzy integrals, we give a weakened definition of generalized triangular norm. For the need of discussion in the following text, we provide and study two conditions (S-1) and (S-2) for generalized triangular norms. Moreover, we enumerate several examples for generalized triangular norms.(2) In chapter 2, we point out that some condition of a convergence theorem of generalized fuzzy integral in [8] (Theorem 1) is not valid by a counterexample. Although we can assure that the sufficiency of this theorem holds under this condition, the necessity is not true. In this paper, We modify it to the right form by weakening the condition. As a corollary of the corrected theorem, under the generalized triangular norm S satisfying (S-1) lim_{α→0}+ S[α, ∞] = 0 condition, we obtain a terse form of convergence theorem of generalized fuzzy integrals, and this result concludes convergence theorem of (S) fuzzy integrals as its particular example.(3) In chapter 3, we study the relation between convergence in fuzzy measure of nonnegative fuzzy measurable functions and fuzzy mean convergence of (G) fuzzy integrals. We give a sufficient and necessary condition where the convergence in fuzzy measure implies the fuzzy mean convergence of (G) fuzzy integrals. The result is an improvement of Theorem 3 in [8]. It shows that the autocontinuity from above assumption of Theorem 3 in [8] is superfluous. Furthermore, as its corollary, we still get several terse and pragmatic sufficiency conditions where the convergence in fuzzy measure implies the fuzzy mean convergence of (G) fuzzy integrals. On the other hand, we also give several terse sufficiency conditions where the fuzzy mean convergence of (G) fuzzy integrals implies the convergence in fuzzy measure and making them equivalent. Similarly, we discuss the relation between fundamental convergence in fuzzy measure and fuzzy mean fundamental convergence of (G) fuzzy integrals. They generalize the corresponding conclusions in (S) fuzzy integral theory.(4) In chapter 4, we discuss further some properties of generalized fuzzy inte-grable functions (sequences), and prove that not only generalized fuzzy integrable functions have the weak integral absolute continuity, but also sequences of generalized fuzzy integrable functions have uniform weak and absolute continuity. Moreover, we still get an sufficient and necessary condition of uniform sequence of generalized fuzzy integrable functions. They improve and generalize the corresponding conclusions on (S) fuzzy integral in [10].