【作者】 任雪昆；

【导师】 吴从炘；

【作者基本信息】 哈尔滨工业大学 ， 基础数学， 2008， 博士

【摘要】 非可加测度理论是数学领域中一个新的研究方向,它与模糊积分为经典测度与积分的拓广,在模糊分析学中占有非常重要的地位,在多目标决策、图像处理、模式识别、人工智能、信息融合和数据挖掘等诸多领域都有重要的应用.因此,进一步开展对非可加测度与模糊Riemann-Stieltjes积分的分析性质方面的研究成为了既有理论意义又有实际价值的研究课题.本文首先研究了单调非可加测度空间中的测度收敛性,然后讨论了非可加测度的空间的拓扑性质,最后研究了模糊Riemann-Stieltjes积分与无穷模糊Riemann-Stieltjes积分的一系列分析性质.本文的主要工作如下:1.举例说明了对于单调非可加测度来说,上自连续性与双零渐近可加性是两个截然不同的概念.通过三个反例说明了上自连续性不能保持测度收敛关于代数运算和格运算的可继承性,并证明了双零渐近可加性则是描述上述性质的充要条件.因而得到:双零渐近可加性是讨论单调非可加测度的测度收敛性的一个非常有效的工具.2.定义了非可加测度的空间中一种新的拓扑—B-拓扑,给出了在这个拓扑意义下的B-收敛,并讨论了该收敛与以往的BV-收敛,B+-收敛之间的关系:μi→BVμ?μi→Bμ?μi→B+μ,以及逆命题成立的充分条件.还运用离散数学的证明方法得到了一个重要的引理:非空集X的σ-代数X必为有限集或不可列集.并在这个引理的基础上得到了(FM(X,X ) ,·)可分当且仅当σ-代数X是有限集,和空间(FM(X,X ) ,·)中的任意依范有界子集恒列紧当且仅当σ?代数X为有限集这两个结论.3.利用一般拓扑学中的一个关于累次极限的定理,修正了关于有限区间上模糊Riemann-Stieltjes积分已有工作中一些重要结论(可积性的充要条件,积分存在定理和积分区间可加定理)证明的错误.还进一步讨论了两类模糊Riemann-Stieltjes积分的运算性质,并首次给出了当模糊数值函数列和实函数列分别收敛时,两类积分的四种积分序列收敛定理.4.基于3,更进一步地研究了无穷模糊Riemann-Stieltjes积分:给出了无穷区间上的两类模糊Riemann-Stieltjes积分的定义;系统地讨论了两类无穷积分的性质,得到了积分的几个运算性质、可积的充要条件、柯西收敛准则、存在性定理以及两类无穷积分的四个积分序列收敛定理等结果.更多还原

【Abstract】 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.更多还原