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Open AccessArticle

On Null-Continuity of Monotone Measures

by Jun Li
School of Sciences, Communication University of China, Beijing 100024, China
Mathematics 2020, 8(2), 205;
Received: 17 January 2020 / Revised: 3 February 2020 / Accepted: 4 February 2020 / Published: 6 February 2020
(This article belongs to the Special Issue Nonlinear Analysis and Optimization)
The null-continuity of monotone measures is a weaker condition than continuity from below and possesses many special properties. This paper further studies this structure characteristic of monotone measures. Some basic properties of null-continuity are shown and the characteristic of null-continuity is described by using convergence of sequence of measurable functions. It is shown that the null-continuity is a necessary condition that the classical Riesz’s theorem remains valid for monotone measures. When considered measurable space ( X , A ) is S-compact, the null-continuity condition is also sufficient for Riesz’s theorem. By means of the equivalence of null-continuity and property (S) of monotone measures, a version of Egoroff’s theorem for monotone measures on S-compact spaces is also presented. We also study the Sugeno integral and the Choquet integral by using null-continuity and generalize some previous results. We show that the monotone measures defined by the Sugeno integral (or the Choquet integral) preserve structural characteristic of null-continuity of the original monotone measures.
Keywords: fuzzy measure; monotone measure; null-continuity; Sugeno integral; Choquet integral; nonlinear integral fuzzy measure; monotone measure; null-continuity; Sugeno integral; Choquet integral; nonlinear integral
MDPI and ACS Style

Li, J. On Null-Continuity of Monotone Measures. Mathematics 2020, 8, 205.

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