On Null-Continuity of Monotone Measures

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 <mml:semantics>(X,A)</mml:semantics> 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.