A Further Investigation for Fuzzy Measures on Metric Spaces

The pseudometric generating property of non-additive set functions also plays an important role as the autocontinuity in fuzzy measure theory [2, 9]. A further research on this property has been presented in [4]. The aim of this paper is to prove the Egoroff's theorem and Lusin's theorem for measurable functions on metric spaces hold for those fuzzy measures with the exhaustivity and pseudometric generating property. Keywords: Fuzzy measure, autocontinuity, exhaustivity, Lusin's theorem, pseudometric generating property, regularity. 1 Introduction The Egoroff's theorem and Lusin's theorem in the classical measure theory are important and useful for discussion of convergence and continuity of measurable functions. The Egoroff's theorem for a fuzzy measure space was first proposed by Wang and Klir [9], but there was supposed the finiteness of fuzzy measures. Further investigations on this matter were discussed by Jiang et al [1, 2, 3] and Ha [7], respectively. However, some stronger co..