The Non-Additive Measure And The Fuzzy Riemann-Stieltjes Integral

The theory of non-additive measure is a new research topic in mathematics. Non-additive measure and fuzzy integral are the extension of classical measure and inte-gral. They occupy an important place in the theory of fuzzy analysis. Application ofthis theory can be found in multicriteria decision making, image processing, patternrecognition, artificial intelligence, information fusing and data mining. Thus, it is atheoretical and practical research topic to study the analytic properties of non-additivemeasure and fuzzy Riemann-Stieltjes integral. In this paper, firstly, the convergencein measure for monotone non-additive measure space is shown. Secondly, some topo-logical properties of the space of non-additive measures is given. Finally, a series ofanalytic properties of fuzzy Riemann-Stieltjes integral and infinite fuzzy Riemann-Stieltjes integral are obtained. The main work of this paper is listed as follows:1. Several examples are given to show that autocontinuity from above is differentfrom double asymptotic null-additivity according to monotone non-additive measure.It is proved that double asymptotic null-additivity is the necessary and sufficient con-dition to keep algebraic operation and lattice operation with respect to convergencein measure, but, for the autocontinuity from above could not preserve these by threeexamples. Therefore, double asymptotic null-additive is a valid tool to study the con-vergence in measure for monotone non-additive measure.2. A new topology– B-topology of the space of non-additive measures isdefined. B-convergence is shown according to B-topology. The relations amongB-convergence, BV-convergence and B+-convergence is obtained:μi→BVμ?μi→Bμ?μi→B+μ, and the sufficient conditions of the converse propositionsare shown. By the proving method in discrete mathematic, an important lemma is ob-tained: theσ-algebra of a nonempty set X is finite set or uncountable set. Accordingto this lemma, the space (FM (X, X ),·) is separable if and only ifσ-algebra Xis finite; in the space (FM (X, X ),·), all bounded subsets with respect to normare compact if and only ifσ-algebra is finite.3. By the theorem on iterated limits in general topology, some mistakes in theproofs of some important conclusions of fuzzy Riemann-Stieltjes integral (the ne- cessity and sufficient condition of integrability, the existence theorem of integral andtheorem of additivity over intervals) in previous work are modified. Then the opera-tion properties of two kinds of fuzzy Riemann-Stieltjes integral are discussed. For thefirst time, four convergence theorems of the sequence of integrals are obtained whenthe sequence of fuzzy number valued functions and the sequence of real functions areconvergent respectively.4. Based on 3, infinite fuzzy Riemann-Stieltjes integral is studied; the definitionof two kinds of fuzzy Riemann-Stieltjes integral on infinite intervals are given; theproperties of the infinite integrals are discussed systematically. We obtained severaloperation properties of the integrals, the necessity and sufficient condition of integra-bility, Cauchy convergence criterion, existence theorem and four convergence theoremof the sequence of integrals.