关于非可加集函数的若干问题的研究
|
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.
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.
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.
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.
引文
1 G. Choquet. Theory of Capacities. Ann. Inst. Fourier. 1954, 5:131–295
2 A. P. Dempster. Upper and Lower Probabilities Induced by Multivalued Map-ping. Annals of Mathematical Statistics. 1967, 38:325–339
3 G. Shafer. A Mathematical Theory of Evidence. Princeton University PressPrinceton, New Jersey, 1976
4 G. Shafer. Belief Function and Possibility Measures. in “ Analysis of FuzzyInformation” (J. C. Bezdek ed.), CRC Press. 1987:51–84
5 L. A. Zadeh. Fuzzy Sets. Information and Control. 1965, 8:338–353
6 M. Sugeno. Theory of Fuzzy Integrals and Its Applications. Ph. D. DissertationTokyo Institute of Technology. 1974
7 P. R. Halmos. Measure Theory. Van Nostrand,New York, 1974
8 D. A. Ralescu, G. Adams. The Fuzzy Integral. J. Math. Anal. Appl. 1980,75:562–570
9 L. A. Zadeh. Fuzzy Sets as a Basis for a Theory of Possibility. Fuzzy Sets andSystems. 1978, 1:3–28
10 王震源. Fuzzy 积分与Fuzzy 卷积. 工程数学学报. 1985, 1:101–110
11 E. P. Klement, W. Schwyhla, R. Lowen. Fuzzy Probability Measures. FuzzySets and Systems. 1981, 5:21–30
12 R. Kruse. On the Construction of Fuzzy Measure. Fuzzy Sets and Systems.1982, 8:323–327
13 R. Kruse. Fuzzy Integrals and Conditional Fuzzy Measures. Fuzzy Sets andSystems. 1983, 10:309–313
14 杨庆季, 宋仁明. Fuzzy 测度空间上泛积分的进一步讨论. 模糊数学. 1985,4:27–36
15 Z. Wang. The Autocontinuity of Set Function and the Fuzzy Integral. J. Math.Anal. Appl. 1984, 99:195–218
16 Z. Wang. On the Null-additivity and the Autocontinuity of a Fuzzy Measure.Fuzzy Sets and Systems. 1992, 45:223–226
17 C. Wu, M. Ha. On the Null-additivity and the Uniform Autocontinuity of aFuzzy Measure. Fuzzy Sets and Systems. 1993, 58:243–245
18 Z. Wang. Asymptotic Structural Characteristics of Fuzzy Measure and TheirApplications. Fuzzy Sets and Systems. 1985, 16:277–290
19 Q. Sun. On the Pseudo-autocontinuity of Fuzzy Measures. Fuzzy Sets andSystems. 1992, 45:59–68
20 哈明虎, 程立新. 关于模糊测度的伪零可加与伪一致自连续性. 河北大学学报 (自然科学版). 1995, 15:13–18
21 S. Weber. ⊥-decomposable Measures and Integrals for Archimedean t-conorm.J. Math. Anal. Appl. 1984, 101:114–138
22 S. Weber. Conditional Measures and Their Applications to Fuzzy Sets. FuzzySets and Systems. 1991, 42:73–85
23 M. Sugeno, T. Murofushi. Pseudo-additive Measures and Integrals. J. Math.Anal. Appl. 1987, 122:197–222
24 G. Deschrijver. Arithmetic Operators in Interval-valued Fuzzy Set Theory. In-formation Sciences. 2007, 177:2906–2924
25 I. Dobrakov. On Submeasures I. Dissertiones Math. 1974, 112:5–35
26 I. Dobrakov, J. Farkova. On Submeasures I I. Math. Slovaca. 1980, 30:65–81
27 N. J. Kalton, J. W. Roberts. Uniformly Exhaustive Submeasures and NearlyAdditive Set Functions. Trans. Amer. Math. Soc. 1989, 278:803–816
28 C. Brook. Decompositions of Submeasures. Canad. J. Math. 1984, 36:577–590
29 E. Pap. A Generalization of the Diagonal Theorem on a Block-matrix. Mat.Vesnik. 1974, 11:66–71
30 S. Saeki. The Vitali - Hahn - Saks Theorem and Measuroids. Proc. Amer. Math.Soc. 1992, 114:775–782
31 E. P. Klement, R. Mesiar, E. Pap. Integration with Respect to Decompos-able Measures Based on a Conditionally Distributive Semiring on the UnitInterval. Internat. J. Uncertain. Fuzziness Knowledge-Based Systems. 2000,8:701–717
32 C. Bertoluzza, V. Doldi. On the Distributivity between t-norms and t-conorms.Fuzzy Sets and Systems. 2004, 142:85–104
33 M. Mas, G. Mayor, J. Torrens. The Modularity Condition for Uninorms andt-operators. Fuzzy Sets and Systems. 2002, 126:207–218
34 M. Mas, G. Mayor, J. Torrens. The Distributivity Condition for Uninorms andt-operators. Fuzzy Sets and Systems. 2002, 128:209–225
35 M. Monserrat, J. Torrens. On the Reversibility of Uninorms and t-operators.Fuzzy Sets and Systems. 2002, 131:303–314
36 M. Mas, G. Mayor, J. Torrens. Corrigendum to “the Distributivity Conditionfor Uninorms and t-operators” [ Fuzzy Sets and Systems, 128 (2002) 209-225].Fuzzy Sets and Systems. 2005, 153:297–299
37 Y. Narukawa, V. Torra. Generalized Transformed t-conorm Integral and Multi-fold Integral. Fuzzy Sets and Systems. 2006, 157:1384–1392
38 W. Hua. The Properties of some Non-additive Measures. Fuzzy Sets and Sys-tems. 1988, 27:373–377
39 V. M. Klimkin. Uniform Boundedness of a Family of Nonadditive Set Functions( Russian). Mat. Sb. 1989, 180:385–396
40 E. Pap. On Non-additive Set Functions. Atti. Sem. Mat. Fis. Univ. Modena.1991, 39:345–360
41 吴从炘, 马明. 模糊分析学基础. 国防工业出版社, 1991
42 Z. Wang, G. J. Klir. Fuzzy Measure Theory. Plenum, New York, 1992
43 D. Denneberg. Non-additive Measure and Integral. Kluwer, Dordrecht, 1994
44 吴从炘, 马明, 方锦暄. 模糊分析学的结构理论. 贵州科技出版社, 1994,1–220
45 Z. Wang, G. J. Klir, D. Harmanec. The Preservation of Structural Character-isitics of Monotone Set Functions Defined by Fuzzy Integral. J. Fuzzy Math.1995, 1:229–240
46 Q. Jiang, H. Suzuki, Z. Wang, et al. Exhaustivity and Absolute Continuity ofFuzzy Measures. Fuzzy Sets and Systems. 1998, 96:231–238
47 哈明虎, 吴从炘. 模糊测度与模糊积分理论. 科学出版社, 1998
48 张强, 刘克. 非可加集函数的Lebesgue分解. 数学学报. 2002, 45:899–904
49 Y. Narukawa, T. Murofushi, M. Sugeno. Space of Fuzzy Measures and Conver-gence. Fuzzy Sets and Systems. 2003, 138:497–506
50 H. Suzuki. On Fuzzy Measures Defined by Fuzzy Integrals. J. Math. Anal.Appl. 1988, 132:87–101
51 Z. Wang, G. Klir, W. Wang. Fuzzy Measures Defined by Fuzzy Integral andTheir Absolute Continuity. J. Math. Anal. Appl. 1996, 203:150–165
52 Z. Wang, G. Klir, W. Wang. Monotone Set Functions Defined by Choquet Inte-gral. Fuzzy Sets and Systems. 1996, 81:241–250
53 Y. Ouyang, J. Li. Some Properties of Monotone Set Function Defined by Cho-quet Integral. Journal of Southeast University. 2003, 19:423–426
54 Y. Ouyang, J. Li. A Note on the Monotone Set Functions Defined by ChoquetIntegral. Fuzzy Sets and Systems. 2004, 146:147–151
55 H. Suzuki. Atoms of Fuzzy Measures and Fuzzy Integrals. Fuzzy Sets andSystems. 1991, 41:329–342
56 Q. Jiang, H. Suzuki. Lebesgue and Saks Decomposition of σ-finite Fuzzy Mea-sures. Fuzzy Sets and Systems. 1995, 75:373–385
57 Q. Jiang, H. Suzuki. The Range of σ-finite Fuzzy Measures. Proc. FUZZ-IEEE/IFES’95 Japan, 1995, 137–144
58 Q. Jiang, H. Suzuki. Fuzzy Measures on Metric Spaces. Fuzzy Sets and Sys-tems. 1996, 83:99–106
59 E. Pap. The Range of Null-additive Fuzzy and Non-fuzzy Measures. FuzzySets and Systems. 1994, 65:105–115
60 C. Wu, C. Wu. A Note on the Range of Null-additive Fuzzy and Non-fuzzyMeasures. Fuzzy Sets and Systems. 2000, 110:145–148
61 王震源. Fuzzy 测度空间上的“几乎”与“伪几乎”的概念. 数学研究与评论.1986, 1:39–41
62 哈明虎, 王熙照. Fuzzy 测度与收敛. 河北大学学报. 1989, 5:79–86
63 哈明虎, 王熙照. 泛可加正规Fuzzy 测度序列的收敛性. 河北师范学院学报. 1991, 4:32–42
64 张德利, 郭彩梅, 王中海. Fuzzy 测度序列的收敛性和积分序列的收敛定理. 东北师范大学学报(自然科学版). 1993, 4:11–14
65 M. Ha, X. Wang. Weak Convergence of Fuzzy Measure Sequences. J. FuzzyMath. 1995, 3:383–391
66 J. Li, M. Yasuda, Q. Jiang, et al. Convergence of Sequence of MeasurableFunctions on Fuzzy Measure Spaces. Fuzzy Sets and Systems. 1997, 87:317–32367 Q. Jiang, S. Wang, D. Ziou. A Further Investigation for Fuzzy Measures onMetric Spaces. Fuzzy Sets and Systems. 1999, 105:293–297
68 J. Li. On Egoroff’s Theorems on the Fuzzy Measure Spaces. Fuzzy Sets andSystems. 2003, 135:367–375
69 J. Li. Order Continuous of Monotone Set Function and Convergence of Mea-surable Functions Sequence. Applied Mathematics and Computation. 2003,135:211–218
70 J. Li. A Further Investigation for Egoroff’s Theorem with Respect to MonotoneSet Functions. Kybernetika. 2003, 39:753–760
71 J. Li, Y. Ouyang, M. Yasuda. Pseudo-convergence of Measurable Functionson Sugeno Fuzzy Measure Spaces. in: Proc. 7th Joint Conf. on InformationScience, North Corolina, USA. 2003:56–59
72 J. Song, J. Li. Regularity of Null-additive Fuzzy Measure on Metric Spaces.International Journal of General Systems. 2003, 32:271–279
73 J. Li, M. Yasuda. Lusin’s Theorem on Fuzzy Measure Spaces. Fuzzy Sets andSystems. 2004, 146:121–133
74 J. Song, J. Li. Lebesgue Theorems in Non-additive Measure Theory. FuzzySets and Systems. 2005, 149:543–548
75 J. Li, M. Yasuda. On Egoroff’s Theorems on Finite Monotone Non-additiveMesaure Spaces. Fuzzy Sets and Systems. 2005, 153:71–78
76 T. Murofushi, K. Uchino, S. Asahina. Conditions for Egoroff’s Theorem inNon-additive Measure Theory. Fuzzy Sets and Systems. 2004, 146:135–146
77 J. Kawabe. The Egoroff Theorem for Non-additive Measures in Riesz Spaces.Fuzzy Sets and Systems. 2006, 157:2762–2770
78 J. Kawabe. The Egoroff Property and the Egoroff Theorem in Riesz Space-valued Non-additive Measure Theory. Fuzzy Sets and Systems. 2007, 158:50–5779 C. Wu, M. Ha. On the Regularity of the Fuzzy Measure on Metric Fuzzy Mea-sure Spaces. Fuzzy Sets and Systems. 1994, 66:373–379
80 M. Ha, X. Wang. Some Notes on the Regularity of Fuzzy Measures on MetricSpaces. Fuzzy Sets and Systems. 1997, 87:385–387
81 J. Wu, C. Wu. Fuzzy Regular Measures on Topogical Spaces. Fuzzy Sets andSystems. 2001, 119:529–533
82 M. Sugeno, Y. Narukawa, T. Murofushi. Choquet Integral and Fuzzy Measureson Locally Compact Space. Fuzzy Sets and Systems. 1998, 99:205–211
83 Y. Narukawa, T. Murofushi. Regular Non-additive Measure and Choquet Inte-gral. Fuzzy Sets and Systems. 2004, 143:487–492
84 J. Li, M. Yasuda, J. Song. Regularity Properties of Null-additive Fuzzy Measureon Metric Spaces. Lecture Notes in Computer Science. 2005, 3558:59–66
85 纪爱兵, 张彦娥, 孙建平. 局部紧空间上的Fuzzy 测度. 模糊系统与数学.1999, 13:36–40
86 王鹏, 刘新平. 局部紧空间上的正则Fuzzy 测度. 陕西师范大学学报(自然科学版). 2001, 29:117–118
87 王亚华, 高遵海. 具有可数基的局部紧Hausdorff 空间上Fuzzy 测度. 武汉工业学院学报. 2004, 23:116–118
88 王震源. 关于Fuzzy 积分序列依测度收敛定理的注记. 模糊数学. 1984,4:7–10
89 Z. Wang. Fuzzy Convolution of Fuzzy Distributions on Group. in “ Analysis ofFuzzy Information” (J. C. Bezdek ed.), CRC Press. 1987:97–103
90 T. Murofushi, M. Sugeno. An Interpretation of Fuzzy Measures and the Cho-quet Integral as an Integral with Respect to a Fuzzy Measure. Fuzzy Sets andSystems. 1989, 29:201–227
91 T. Murofushi, M. Sugeno. Fuzzy t-conorm Integral with Respect to Fuzzy Mea-sures: Generalization of Sugeno Integral and Choquet Integral. Fuzzy Sets andSystems. 1991, 42:57–71
92 T. Murofushi, M. Sugeno. A Theory of Fuzzy Measures: Representations, theChoquet Integral, and Null Sets. J. Math. Anal. Appl. 1991, 159:532–549
93 T. Murofushi, M. Sugeno. Non-monotonic Fuzzy Measures and the ChoquetIntegral. Fuzzy Sets and Systems. 1994, 64:73–86
94 A. D. Waegenaere, R. Kast, A. Lapied. Choquet Pricing and Equilibrium. In-surance: Mathematics and Economics. 2003, 32:359–370
95 Y. Narukawa, T. Murofushi. Conditions for Choquet Integral Representation ofthe Comonotonically Additive and Monotone Functional. J. Math. Anal. Appl.2003, 282:201–211
96 C. Guo, D. Zhang. Fuzzy-valued Choquet Integrals (Ⅱ)-the Choquet Integralof Functions with Respect to Fuzzy-valued Fuzzy Measures. 模糊系统与数学. 2003, 17:23–28
97 Y. Narukawa, T. Murofushi. Choquet Integral with Respect to a Regular Non-additive Measure. IEEE International Conference on Fuzzy Systems (Budapest,Hungary). 2004:25–29
98 Y. Rebille. Sequentially Continuous Non-monotonic Choquet Integrals. FuzzySets and Systems. 2005, 153:79–94
99 张德利, 郭彩梅, 吴从炘. 模糊积分论进展. 模糊系统与数学. 2003,17:1–10
100 Y. W. Chen, G. H. Tzeng. Using Fuzzy Integral for Evaluating SubjectivelyPerceived Travel Costs in a Traffic Assignment Model. European Journal ofOperational Research. 2001, 130:653–664
101 Y. W. Chen, H. L. Chang, G. H. Tzeng. Using Fuzzy Measures and Habit-ual Domains to Analyze the Public Attitude and Apply to the Gas Taxi Policy.European Journal of Operational Research. 2002, 137:145–161
102 K. Leszczynski, P. Penczek, W. Grochuiski. Sugeno’s Fuzzy Measure and FuzzyClustering. Fuzzy Sets and Systems. 1985, 15:147–158
103 T. Onisava. Fuzzy Measure Analysis of Publicattitude Towards the Use of Nu-clear Energy. Fuzzy Sets and Systems. 1986, 20:259–289
104 李荣钧. 模糊多准则决策理论与应用. 科学出版社, 2002
105 彭祖赠, 孙韫玉. 模糊数学及其应用. 武汉大学出版社, 2002
106 胡宝清. 模糊理论基础. 武汉大学出版社, 2004, 395–434
107 赵汝怀. ( N) 模糊积分. 数学研究与评论. 1981, 2:55–72
108 杨庆季. Fuzzy 测度空间上的泛积分. 模糊数学. 1985, 3:107–114
109 Z. Wang. Une Classe De Measures Floues-les Quasi-measure. BUSEFAL.1981, 6:28–37
110 王震源. gλ型Fuzzy测度的构造与拟概率. 河北大学学报. 1983, 1:79–83
111 宋仁明. 一类模糊测度的扩张. 河北大学学报. 1984, 2:97–101
112 Z. Wang, Z. Zhang. On the Extension of Possibility Measures. BUSEFAL.1984, 18:26–32
113 王震源, 张志鹏. 广义可能性测度扩张定理. 科学通报. 1984, 15:959
114 Z. Wang. Extension of Possibility Measures Defined on an Arbitrary NonemptyClass of Sets. Proc. IFSA’85, Palma de Mallorca. 1985
115 G. Banon. Distinction between Several Subsets of Fuzzy Measures. Fuzzy Setsand Systems. 1981, 5:291–305
116 刘贵廷, 王震源. 一致信任函数扩张定理. 科学通报. 1985, 9:718–719
117 Z. Wang. Extension of Consonant Belief Functions Defined on an ArbitraryNonempty Class of Sets. Publication 54 de la Groupe de Recherche ClaudeFrancois Picard, C. N. R. S., France. 1985:61–65
118 E. Pap. Extension of the Continuous t-conorm Decomposable Measure. Univ.u Novom Sadu Zb. Rad. Prirod.-Mat.Fak. Ser. Mat. 1990, 20:121–130
119 E. Pap. Extension of Null-additive Set Functions on Algebra of Subsets. NoviSad J. Math. 2001, 31:9–13
120 白山. 非可加测度的自生成测度. 山东大学学报(理学版). 2004, 39:28–33
121 E. Pap. Null-additive Set Function. Kluwer, Dordrecht, 1995
122 S. Rao, M. Rao. Theory of Charges. Academic Press, 1980
2 A. P. Dempster. Upper and Lower Probabilities Induced by Multivalued Map-ping. Annals of Mathematical Statistics. 1967, 38:325–339
3 G. Shafer. A Mathematical Theory of Evidence. Princeton University PressPrinceton, New Jersey, 1976
4 G. Shafer. Belief Function and Possibility Measures. in “ Analysis of FuzzyInformation” (J. C. Bezdek ed.), CRC Press. 1987:51–84
5 L. A. Zadeh. Fuzzy Sets. Information and Control. 1965, 8:338–353
6 M. Sugeno. Theory of Fuzzy Integrals and Its Applications. Ph. D. DissertationTokyo Institute of Technology. 1974
7 P. R. Halmos. Measure Theory. Van Nostrand,New York, 1974
8 D. A. Ralescu, G. Adams. The Fuzzy Integral. J. Math. Anal. Appl. 1980,75:562–570
9 L. A. Zadeh. Fuzzy Sets as a Basis for a Theory of Possibility. Fuzzy Sets andSystems. 1978, 1:3–28
10 王震源. Fuzzy 积分与Fuzzy 卷积. 工程数学学报. 1985, 1:101–110
11 E. P. Klement, W. Schwyhla, R. Lowen. Fuzzy Probability Measures. FuzzySets and Systems. 1981, 5:21–30
12 R. Kruse. On the Construction of Fuzzy Measure. Fuzzy Sets and Systems.1982, 8:323–327
13 R. Kruse. Fuzzy Integrals and Conditional Fuzzy Measures. Fuzzy Sets andSystems. 1983, 10:309–313
14 杨庆季, 宋仁明. Fuzzy 测度空间上泛积分的进一步讨论. 模糊数学. 1985,4:27–36
15 Z. Wang. The Autocontinuity of Set Function and the Fuzzy Integral. J. Math.Anal. Appl. 1984, 99:195–218
16 Z. Wang. On the Null-additivity and the Autocontinuity of a Fuzzy Measure.Fuzzy Sets and Systems. 1992, 45:223–226
17 C. Wu, M. Ha. On the Null-additivity and the Uniform Autocontinuity of aFuzzy Measure. Fuzzy Sets and Systems. 1993, 58:243–245
18 Z. Wang. Asymptotic Structural Characteristics of Fuzzy Measure and TheirApplications. Fuzzy Sets and Systems. 1985, 16:277–290
19 Q. Sun. On the Pseudo-autocontinuity of Fuzzy Measures. Fuzzy Sets andSystems. 1992, 45:59–68
20 哈明虎, 程立新. 关于模糊测度的伪零可加与伪一致自连续性. 河北大学学报 (自然科学版). 1995, 15:13–18
21 S. Weber. ⊥-decomposable Measures and Integrals for Archimedean t-conorm.J. Math. Anal. Appl. 1984, 101:114–138
22 S. Weber. Conditional Measures and Their Applications to Fuzzy Sets. FuzzySets and Systems. 1991, 42:73–85
23 M. Sugeno, T. Murofushi. Pseudo-additive Measures and Integrals. J. Math.Anal. Appl. 1987, 122:197–222
24 G. Deschrijver. Arithmetic Operators in Interval-valued Fuzzy Set Theory. In-formation Sciences. 2007, 177:2906–2924
25 I. Dobrakov. On Submeasures I. Dissertiones Math. 1974, 112:5–35
26 I. Dobrakov, J. Farkova. On Submeasures I I. Math. Slovaca. 1980, 30:65–81
27 N. J. Kalton, J. W. Roberts. Uniformly Exhaustive Submeasures and NearlyAdditive Set Functions. Trans. Amer. Math. Soc. 1989, 278:803–816
28 C. Brook. Decompositions of Submeasures. Canad. J. Math. 1984, 36:577–590
29 E. Pap. A Generalization of the Diagonal Theorem on a Block-matrix. Mat.Vesnik. 1974, 11:66–71
30 S. Saeki. The Vitali - Hahn - Saks Theorem and Measuroids. Proc. Amer. Math.Soc. 1992, 114:775–782
31 E. P. Klement, R. Mesiar, E. Pap. Integration with Respect to Decompos-able Measures Based on a Conditionally Distributive Semiring on the UnitInterval. Internat. J. Uncertain. Fuzziness Knowledge-Based Systems. 2000,8:701–717
32 C. Bertoluzza, V. Doldi. On the Distributivity between t-norms and t-conorms.Fuzzy Sets and Systems. 2004, 142:85–104
33 M. Mas, G. Mayor, J. Torrens. The Modularity Condition for Uninorms andt-operators. Fuzzy Sets and Systems. 2002, 126:207–218
34 M. Mas, G. Mayor, J. Torrens. The Distributivity Condition for Uninorms andt-operators. Fuzzy Sets and Systems. 2002, 128:209–225
35 M. Monserrat, J. Torrens. On the Reversibility of Uninorms and t-operators.Fuzzy Sets and Systems. 2002, 131:303–314
36 M. Mas, G. Mayor, J. Torrens. Corrigendum to “the Distributivity Conditionfor Uninorms and t-operators” [ Fuzzy Sets and Systems, 128 (2002) 209-225].Fuzzy Sets and Systems. 2005, 153:297–299
37 Y. Narukawa, V. Torra. Generalized Transformed t-conorm Integral and Multi-fold Integral. Fuzzy Sets and Systems. 2006, 157:1384–1392
38 W. Hua. The Properties of some Non-additive Measures. Fuzzy Sets and Sys-tems. 1988, 27:373–377
39 V. M. Klimkin. Uniform Boundedness of a Family of Nonadditive Set Functions( Russian). Mat. Sb. 1989, 180:385–396
40 E. Pap. On Non-additive Set Functions. Atti. Sem. Mat. Fis. Univ. Modena.1991, 39:345–360
41 吴从炘, 马明. 模糊分析学基础. 国防工业出版社, 1991
42 Z. Wang, G. J. Klir. Fuzzy Measure Theory. Plenum, New York, 1992
43 D. Denneberg. Non-additive Measure and Integral. Kluwer, Dordrecht, 1994
44 吴从炘, 马明, 方锦暄. 模糊分析学的结构理论. 贵州科技出版社, 1994,1–220
45 Z. Wang, G. J. Klir, D. Harmanec. The Preservation of Structural Character-isitics of Monotone Set Functions Defined by Fuzzy Integral. J. Fuzzy Math.1995, 1:229–240
46 Q. Jiang, H. Suzuki, Z. Wang, et al. Exhaustivity and Absolute Continuity ofFuzzy Measures. Fuzzy Sets and Systems. 1998, 96:231–238
47 哈明虎, 吴从炘. 模糊测度与模糊积分理论. 科学出版社, 1998
48 张强, 刘克. 非可加集函数的Lebesgue分解. 数学学报. 2002, 45:899–904
49 Y. Narukawa, T. Murofushi, M. Sugeno. Space of Fuzzy Measures and Conver-gence. Fuzzy Sets and Systems. 2003, 138:497–506
50 H. Suzuki. On Fuzzy Measures Defined by Fuzzy Integrals. J. Math. Anal.Appl. 1988, 132:87–101
51 Z. Wang, G. Klir, W. Wang. Fuzzy Measures Defined by Fuzzy Integral andTheir Absolute Continuity. J. Math. Anal. Appl. 1996, 203:150–165
52 Z. Wang, G. Klir, W. Wang. Monotone Set Functions Defined by Choquet Inte-gral. Fuzzy Sets and Systems. 1996, 81:241–250
53 Y. Ouyang, J. Li. Some Properties of Monotone Set Function Defined by Cho-quet Integral. Journal of Southeast University. 2003, 19:423–426
54 Y. Ouyang, J. Li. A Note on the Monotone Set Functions Defined by ChoquetIntegral. Fuzzy Sets and Systems. 2004, 146:147–151
55 H. Suzuki. Atoms of Fuzzy Measures and Fuzzy Integrals. Fuzzy Sets andSystems. 1991, 41:329–342
56 Q. Jiang, H. Suzuki. Lebesgue and Saks Decomposition of σ-finite Fuzzy Mea-sures. Fuzzy Sets and Systems. 1995, 75:373–385
57 Q. Jiang, H. Suzuki. The Range of σ-finite Fuzzy Measures. Proc. FUZZ-IEEE/IFES’95 Japan, 1995, 137–144
58 Q. Jiang, H. Suzuki. Fuzzy Measures on Metric Spaces. Fuzzy Sets and Sys-tems. 1996, 83:99–106
59 E. Pap. The Range of Null-additive Fuzzy and Non-fuzzy Measures. FuzzySets and Systems. 1994, 65:105–115
60 C. Wu, C. Wu. A Note on the Range of Null-additive Fuzzy and Non-fuzzyMeasures. Fuzzy Sets and Systems. 2000, 110:145–148
61 王震源. Fuzzy 测度空间上的“几乎”与“伪几乎”的概念. 数学研究与评论.1986, 1:39–41
62 哈明虎, 王熙照. Fuzzy 测度与收敛. 河北大学学报. 1989, 5:79–86
63 哈明虎, 王熙照. 泛可加正规Fuzzy 测度序列的收敛性. 河北师范学院学报. 1991, 4:32–42
64 张德利, 郭彩梅, 王中海. Fuzzy 测度序列的收敛性和积分序列的收敛定理. 东北师范大学学报(自然科学版). 1993, 4:11–14
65 M. Ha, X. Wang. Weak Convergence of Fuzzy Measure Sequences. J. FuzzyMath. 1995, 3:383–391
66 J. Li, M. Yasuda, Q. Jiang, et al. Convergence of Sequence of MeasurableFunctions on Fuzzy Measure Spaces. Fuzzy Sets and Systems. 1997, 87:317–32367 Q. Jiang, S. Wang, D. Ziou. A Further Investigation for Fuzzy Measures onMetric Spaces. Fuzzy Sets and Systems. 1999, 105:293–297
68 J. Li. On Egoroff’s Theorems on the Fuzzy Measure Spaces. Fuzzy Sets andSystems. 2003, 135:367–375
69 J. Li. Order Continuous of Monotone Set Function and Convergence of Mea-surable Functions Sequence. Applied Mathematics and Computation. 2003,135:211–218
70 J. Li. A Further Investigation for Egoroff’s Theorem with Respect to MonotoneSet Functions. Kybernetika. 2003, 39:753–760
71 J. Li, Y. Ouyang, M. Yasuda. Pseudo-convergence of Measurable Functionson Sugeno Fuzzy Measure Spaces. in: Proc. 7th Joint Conf. on InformationScience, North Corolina, USA. 2003:56–59
72 J. Song, J. Li. Regularity of Null-additive Fuzzy Measure on Metric Spaces.International Journal of General Systems. 2003, 32:271–279
73 J. Li, M. Yasuda. Lusin’s Theorem on Fuzzy Measure Spaces. Fuzzy Sets andSystems. 2004, 146:121–133
74 J. Song, J. Li. Lebesgue Theorems in Non-additive Measure Theory. FuzzySets and Systems. 2005, 149:543–548
75 J. Li, M. Yasuda. On Egoroff’s Theorems on Finite Monotone Non-additiveMesaure Spaces. Fuzzy Sets and Systems. 2005, 153:71–78
76 T. Murofushi, K. Uchino, S. Asahina. Conditions for Egoroff’s Theorem inNon-additive Measure Theory. Fuzzy Sets and Systems. 2004, 146:135–146
77 J. Kawabe. The Egoroff Theorem for Non-additive Measures in Riesz Spaces.Fuzzy Sets and Systems. 2006, 157:2762–2770
78 J. Kawabe. The Egoroff Property and the Egoroff Theorem in Riesz Space-valued Non-additive Measure Theory. Fuzzy Sets and Systems. 2007, 158:50–5779 C. Wu, M. Ha. On the Regularity of the Fuzzy Measure on Metric Fuzzy Mea-sure Spaces. Fuzzy Sets and Systems. 1994, 66:373–379
80 M. Ha, X. Wang. Some Notes on the Regularity of Fuzzy Measures on MetricSpaces. Fuzzy Sets and Systems. 1997, 87:385–387
81 J. Wu, C. Wu. Fuzzy Regular Measures on Topogical Spaces. Fuzzy Sets andSystems. 2001, 119:529–533
82 M. Sugeno, Y. Narukawa, T. Murofushi. Choquet Integral and Fuzzy Measureson Locally Compact Space. Fuzzy Sets and Systems. 1998, 99:205–211
83 Y. Narukawa, T. Murofushi. Regular Non-additive Measure and Choquet Inte-gral. Fuzzy Sets and Systems. 2004, 143:487–492
84 J. Li, M. Yasuda, J. Song. Regularity Properties of Null-additive Fuzzy Measureon Metric Spaces. Lecture Notes in Computer Science. 2005, 3558:59–66
85 纪爱兵, 张彦娥, 孙建平. 局部紧空间上的Fuzzy 测度. 模糊系统与数学.1999, 13:36–40
86 王鹏, 刘新平. 局部紧空间上的正则Fuzzy 测度. 陕西师范大学学报(自然科学版). 2001, 29:117–118
87 王亚华, 高遵海. 具有可数基的局部紧Hausdorff 空间上Fuzzy 测度. 武汉工业学院学报. 2004, 23:116–118
88 王震源. 关于Fuzzy 积分序列依测度收敛定理的注记. 模糊数学. 1984,4:7–10
89 Z. Wang. Fuzzy Convolution of Fuzzy Distributions on Group. in “ Analysis ofFuzzy Information” (J. C. Bezdek ed.), CRC Press. 1987:97–103
90 T. Murofushi, M. Sugeno. An Interpretation of Fuzzy Measures and the Cho-quet Integral as an Integral with Respect to a Fuzzy Measure. Fuzzy Sets andSystems. 1989, 29:201–227
91 T. Murofushi, M. Sugeno. Fuzzy t-conorm Integral with Respect to Fuzzy Mea-sures: Generalization of Sugeno Integral and Choquet Integral. Fuzzy Sets andSystems. 1991, 42:57–71
92 T. Murofushi, M. Sugeno. A Theory of Fuzzy Measures: Representations, theChoquet Integral, and Null Sets. J. Math. Anal. Appl. 1991, 159:532–549
93 T. Murofushi, M. Sugeno. Non-monotonic Fuzzy Measures and the ChoquetIntegral. Fuzzy Sets and Systems. 1994, 64:73–86
94 A. D. Waegenaere, R. Kast, A. Lapied. Choquet Pricing and Equilibrium. In-surance: Mathematics and Economics. 2003, 32:359–370
95 Y. Narukawa, T. Murofushi. Conditions for Choquet Integral Representation ofthe Comonotonically Additive and Monotone Functional. J. Math. Anal. Appl.2003, 282:201–211
96 C. Guo, D. Zhang. Fuzzy-valued Choquet Integrals (Ⅱ)-the Choquet Integralof Functions with Respect to Fuzzy-valued Fuzzy Measures. 模糊系统与数学. 2003, 17:23–28
97 Y. Narukawa, T. Murofushi. Choquet Integral with Respect to a Regular Non-additive Measure. IEEE International Conference on Fuzzy Systems (Budapest,Hungary). 2004:25–29
98 Y. Rebille. Sequentially Continuous Non-monotonic Choquet Integrals. FuzzySets and Systems. 2005, 153:79–94
99 张德利, 郭彩梅, 吴从炘. 模糊积分论进展. 模糊系统与数学. 2003,17:1–10
100 Y. W. Chen, G. H. Tzeng. Using Fuzzy Integral for Evaluating SubjectivelyPerceived Travel Costs in a Traffic Assignment Model. European Journal ofOperational Research. 2001, 130:653–664
101 Y. W. Chen, H. L. Chang, G. H. Tzeng. Using Fuzzy Measures and Habit-ual Domains to Analyze the Public Attitude and Apply to the Gas Taxi Policy.European Journal of Operational Research. 2002, 137:145–161
102 K. Leszczynski, P. Penczek, W. Grochuiski. Sugeno’s Fuzzy Measure and FuzzyClustering. Fuzzy Sets and Systems. 1985, 15:147–158
103 T. Onisava. Fuzzy Measure Analysis of Publicattitude Towards the Use of Nu-clear Energy. Fuzzy Sets and Systems. 1986, 20:259–289
104 李荣钧. 模糊多准则决策理论与应用. 科学出版社, 2002
105 彭祖赠, 孙韫玉. 模糊数学及其应用. 武汉大学出版社, 2002
106 胡宝清. 模糊理论基础. 武汉大学出版社, 2004, 395–434
107 赵汝怀. ( N) 模糊积分. 数学研究与评论. 1981, 2:55–72
108 杨庆季. Fuzzy 测度空间上的泛积分. 模糊数学. 1985, 3:107–114
109 Z. Wang. Une Classe De Measures Floues-les Quasi-measure. BUSEFAL.1981, 6:28–37
110 王震源. gλ型Fuzzy测度的构造与拟概率. 河北大学学报. 1983, 1:79–83
111 宋仁明. 一类模糊测度的扩张. 河北大学学报. 1984, 2:97–101
112 Z. Wang, Z. Zhang. On the Extension of Possibility Measures. BUSEFAL.1984, 18:26–32
113 王震源, 张志鹏. 广义可能性测度扩张定理. 科学通报. 1984, 15:959
114 Z. Wang. Extension of Possibility Measures Defined on an Arbitrary NonemptyClass of Sets. Proc. IFSA’85, Palma de Mallorca. 1985
115 G. Banon. Distinction between Several Subsets of Fuzzy Measures. Fuzzy Setsand Systems. 1981, 5:291–305
116 刘贵廷, 王震源. 一致信任函数扩张定理. 科学通报. 1985, 9:718–719
117 Z. Wang. Extension of Consonant Belief Functions Defined on an ArbitraryNonempty Class of Sets. Publication 54 de la Groupe de Recherche ClaudeFrancois Picard, C. N. R. S., France. 1985:61–65
118 E. Pap. Extension of the Continuous t-conorm Decomposable Measure. Univ.u Novom Sadu Zb. Rad. Prirod.-Mat.Fak. Ser. Mat. 1990, 20:121–130
119 E. Pap. Extension of Null-additive Set Functions on Algebra of Subsets. NoviSad J. Math. 2001, 31:9–13
120 白山. 非可加测度的自生成测度. 山东大学学报(理学版). 2004, 39:28–33
121 E. Pap. Null-additive Set Function. Kluwer, Dordrecht, 1995
122 S. Rao, M. Rao. Theory of Charges. Academic Press, 1980