Regularity of upper semicontinuous fuzzy measures

In this note, a kind of regularity of fuzzy measures is discussed by using weakly null-additivity of set function and an equivalence condition for Egoroff´s theorem. A version of Egoroff´s theorem and Lusin´s theorem for upper semicontinuous fuzzy measures on a metric space is shown, respectively. As an application of regularity and Lusin´s theorem, the mean approximations of measurable function by continuous in the sense of the Sugeno integral and of the Choquet integral are presented.