The aim of the present paper is to study the regularity and tightness
properties of fuzzy measures on metric spaces. Among other results, we
shall give Lusin's theorem and Egoroff's theorem; we shall also prove
the singleton characterization of atom of fuzzy measure, and an algor
ithm to evaluate the fuzzy integrals of non-negative measurable functi
ons over an atom and a union of finite disjoint atoms.