Applications of null-additive set functions in mathematics. (English) Zbl 0968.28010

Skorokhod, A. V. (ed.) et al., Exploring stochastic laws. Festschrift in honour of the 70th birthday of Academician Vladimir Semenovich Korolyuk. Utrecht: VSP. 367-382 (1995).
The paper gives a survey about some classes of null-additive set functions and its applications. A function \(m: R\to [0,+\infty)\) on a ring \(R\) is called null-additive if \(m(A\cup B)= m(A)\) whenever \(A,B\in R\), \(A\cap B=\emptyset\) and \(\mu(B)= 0\). Examples for null-additive set functions are submeasures and triangular functions. A function \(m: R\to [0,+\infty[\) is said to be triangular if \(m(\emptyset)= 0\) and \(m(A)- m(B)\leq m(A\cup B)\leq m(A)+ m(B)\) whenever \(A,B\in R\) and \(A\cap B= \emptyset\). The paper presents a diagonal theorem for triangular set functions and several applications of this diagonal theorem, in particular Nikodým convergence theorem, Nikodým boundedness theorem, Orlicz-Pettis theorem, theorems about continuous convergence, equicontinuity, weak boundedness and about weak\(^*\)-boundedness. In the last section it is shown by examples how the “\(g\)-calculus” can be used in solving differential equations and difference equations.
For the entire collection see [Zbl 0942.60001].
Reviewer: Hans Weber (Udine)


28E10 Fuzzy measure theory