Regularity of fuzzy measures on complete and separable metric spaces

We shall study the regularity of continuous monotone non-additive measures defined on complete separable metric spaces. We prove that a finite continuous non-additive measures is tight, and furthermore, if it is weakly null-additive, then it possesses regularity. By using the regularity, we shall improve and generalized the Lusin theorem and the Egoroff theorem for continuous non-additive measures on metric spaces. We also show the mean approximation theorems in the sense of the Choquet integral and fuzzy integral, respectively. Some properties of atoms of regular non-additive measures are studied, and their characteristics on fuzzy integrals and the Choquet integrals are described.