A new concept of the fuzzy mean fundamental convergence is introduced, and the relations among several convergences ofsequences of(S) fuzzy integrable functions are discussed. Further, the equivalent relation between fuzzy mean fundamental convergence and fundamental convergence in fuzzy measure of sequences of (S) fuzzy integrable functions is proved. Three new definitions of uniform sequence, uniform weak and absolute continuous sequence, and uniform bounded sequence of (S) fuzzy integrable functions are also given, and the properties ofsequences of(S) fuzzy integrable functions are discussed. Moreover, the equivalent relation between the uniform sequence and the uniform bounded sequence of (S) fuzzy integrable functions is shown.