Affiliations:
School of Sciences, Communication University of China, Beijing, China
Correspondence:
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Corresponding author. Jun Li, School of Sciences, Communication University of China, Beijing 100024, China. Tel./Fax: +86 10 65783583; E-mail: lijun@cuc.edu.cn.
Abstract: The regularity of continuous fuzzy measures on complete separable metric spaces are discussed. It is shown that a finite continuous fuzzy measure is tight on complete and separable metric spaces, and if the fuzzy measure is weakly null-additive, then it is regular. A version of Egoroff’s theorem and Lusin’s theorem for fuzzy measures on complete separable metric spaces is shown, respectively. As an application of regularity and Lusin’s theorem, the mean approximation of measurable function by continuous functions in the sense of the Sugeno integral and of the Choquet integral are shown, while a pan-approximation theorem of regular fuzzy neural networks in the sense of the Sugeno integral norm is also presented. The Sugeno integral and the Choquet integral over an atom of a fuzzy measure are characterized.
