Locally and globally small Riemann sums and Henstock integral of fuzzy-number-valued functions

摘要

In this paper, we first define and discuss the locally small Riemann sums (LSRS) for fuzzy-numbervalued functions. In addition the necessary and suffcient conditions have been obtained for a fuzzy-number-valued function which has (LSRS), i.e., if a fuzzy-number-valued function is Henstock (H) integrable on [a, b] then it has (LSRS) and the converse is always true. Secondly, the globally small Riemann sums (GSRS) for fuzzynumber-valued functions is defined and discussed, and the necessary and suffcient conditions have been given for a fuzzy-number-valued function which has (GSRS), i.e., if a fuzzy-number-valued function is (H) integrable on [a, b] then it has (GSRS) and the converse is always true. Finally, by Egorov,s Theorem, we obtain the dominated convergence theorem for globally small Riemann sums (GSRS) of fuzzy-number-valued functions.