In this paper we consider the fuzzy-valued integrals of fuzzy-valued measurable functions with respect to fuzzy-valued measures. We invoke the Hausdorff metric to define the metric between two fuzzy numbers, and then to consider the limit of a sequence of fuzzy numbers. In the meantime, we define the supremum and infimum of a sequence of fuzzy numbers. In this setting, the meaning of convergence of a sequence of fuzzy-valued integrals becomes clear; thus we can prove the Monotone Convergence Theorem, Fatou's Lemma and the Dominated Convergence Theorem. Furthermore, the α-level set of this fuzzy-valued integral is a closed interval whose end points are the classical Lebesgue integrals, which are more tractable.
Fuzzy Sets and Systems / Volume 87, Issue 1, P.65–78