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.
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