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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)

##### MSC:
 2.8e+11 Fuzzy measure theory