Summary: A kind of regularity of monotone measures is shown by using the pseudometric generating property of set function (for short (p.g.p.)) and an equivalence condition for Egoroff’s theorem. Lusin’s theorem, which is well-known in classical measure theory, is generalized to monotone measure spaces. The continuity from above and below of set functions is not required in the discussions; therefore, the previous results obtained by Q. Jiang
et al. [Fuzzy Sets Syst. 83, No. 1, 99–106 (1996; Zbl 0878.28013
); ibid. 105, No. 2, 293–297 (1999; Zbl 0954.28009
)] are generalized. An approximation of measurable functions by continuous functions on monotone measure spaces is presented.