【作者】 孙波；

【导师】 吴从炘；

【作者基本信息】 哈尔滨工业大学 ， 基础数学， 2007， 博士

【摘要】 非可加集函数理论是一个新的数学分支,是经典测度理论的推广,在知识工程,人工智能,博弈论,统计学,经济学和社会学等领域都有重要的应用.所以,研究独立于它的应用的关于非可加集函数的统一的数学理论成为既有理论意义又有实际价值的研究课题.本文首先研究了非可加集函数的原子和伪原子,然后讨论了子集代数上零可加集函数和双零可加集函数的扩张问题,最后研究了局部紧Hausdorff空间上的模糊测度的正则性.本文主要工作如下:1.举例说明了Pap的关于零可加集函数的原子的一篇论文中的引理1(也就是《Null-additive Set Function》一书中的引理6.4)的充分性是不成立的,同时给出了该问题的正确的充分必要条件和证明.2.针对前人都是在零可加性条件下研究非可加集函数的原子的情况,本文讨论了非零可加条件下的非可加集函数的原子,得到了许多与零可加性条件下相同的结论(例如Saks分解定理).3.我们定义了非可加集函数的伪原子,通过举例说明了非可加集函数的原子和伪原子的不同与联系,同时指出,我们可以把所有的原子和伪原子分为三种情形:情形I是伪原子但不是原子;情形II是原子但不是伪原子;情形III既是原子也是伪原子.特别指出,双零可加性条件可能是非可加集函数的伪原子的最佳研究环境.给出了关于非可加集函数的伪原子的性质和定理,它们在形式上与有关非可加集函数的原子的相应结论类似,但实质是不同的.证明了零可加意义下的关于非可加集函数的原子和伪原子的分解定理.4.举例说明了Pap的关于子集代数上零可加集函数扩张问题的一篇论文中的定理2的证明是错误的,同时给出了正确的证明,进一步地指出这种扩张保持单调性.仿照零可加集函数的扩张定理,我们又给出了满足单调性的双零可加集函数的扩张定理,并指出该扩张也可以保持单调性.5.给出了局部紧Hausdorff空间X上的内(外)正则集,正则集以及正则模糊测度的定义.得到了模糊测度正则的充要条件和任意两个紧(或紧Gδ)集的正常差为内(外)正则集的条件,同时证明了单调递增的内正则集的并是内正则的以及具有有限模糊测度的单调递减的外正则集的交是外正则的.此外,在严格单调条件下,我们还证明了具有有限模糊测度的有限个两两不交内正则集的并是内正则的以及每一个紧(或紧Gδ)集是外正则的当且仅当每一个有界开集是内正则的.更多还原

【Abstract】 The theory of non-additive set functions is a new branch of mathematics. As ageneralization of the classical measure theory, non-additive set functions have beenapplied in many areas, such as knowledge engineering, artificial intelligence, gametheory, statistics, economy and sociology. Therefore, the study of non-additive setfunctions has important value of theory and application. Atoms and pseudo-atomsare studied firstly in this paper. Then we discuss the extensions of null-additive setfunctions and null-null-additive set functions on algebra of subsets. At last, we discussthe regularity of fuzzy measure on locally compact Hausdorff spaces. Our main workis as following:1. We point out that the sufficiency of Lemma 1 about null-additive set func-tions in Pap’s paper (i.e. Lemma 6.4 in monograph“Null-additive Set Function”) isincorrect by giving a counterexample. At the same time, we give the sufficient andnecessary condition of this problem and its proof.2. Since a lot of studies on atoms of non-additive set functions under null-additivity condition have been done, we discuss the properties of atoms of non-additive set functions under non-null-additivity condition, and obtain some results(such as Saks decomposition theorem) similar to null-additive set functions.3. We introduce the definition of pseudo-atoms of non-additive set functions andgive the relation between atoms and pseudo-atoms by the method of illustration. Wealso point out that all atoms and pseudo-atoms can be divided into three classes: classI is the set of all pseudo-atoms which are not atoms; class II is the set of all atomswhich are not pseudo-atoms; class III is the set of all atoms that are also pseudo-atoms. Specially, we point out that null-null-additivity may be an adequate framefor pseudo-atoms of non-additive set functions. We give some properties of thesepseudo-atoms which are similar to atoms of non-additive set functions as a matter ofform. Essentially, they are different. We prove a decompose theorem about atoms andpseudo-atoms under the null-additivity condition.4. We point out that the proof of Theorem 2 in Pap’s paper about the extension ofnull-additive set functions on algebra of subsets is wrong by giving a counterexample and correct the old proof. We further show that this extension preserves monotonicity.Similar to the extension theorem of null-additive set functions, we give the extensiontheorem of null-null-additive set functions which satisfies monotonicity. At the sametime, we show that this extension also can keep monotonicity.5. We give the concepts of inner (outer) regular set, regular set and regular fuzzymeasure on locally compact Hausdorff space X. We obtain a necessary and sufficientcondition that a fuzzy measure is regular. At the same time, we show that the sufficientcondition that every proper difference of two compact (or compact Gδ) sets is inner(outer) regular. We show that the union of an increasing sequence of inner regularsets is inner regular, and the intersection of outer regular sets of finite measure is outerregular. For strictly monotone fuzzy measure, we show that the finite disjoint unionof inner regular sets of finite measure is inner regular and every compact (or compactGδ) set is outer regular, if and only if every bounded open set is inner regular.更多还原