区间上单调递减函数的Denneberg积分和Lebesgue积分的等价性
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摘要
为了使得非可加测度和积分理论有更广泛的适用性,结合实分析的方法,通过把取值为负的递减函数转化为非负函数,证明了Denneberg利用分布函数引入的关于区间上单调递减函数的积分与Lebesgue积分恒等价;研究了Denneberg积分的分析性质,给出了几类收敛定理如单调收敛定理、有界收敛定理、控制收敛定理等,从而为该积分的研究提供更多的方法。
In order to make the non-additive measure and integral theory more widely applied, the real analysis method is used to transform the decreasing function with negative value as non-negative function so as to prove that the integral of monotone decreasing function on the interval introduced by Denneberg with distribution function is equivalent to the Lebesgue integral. Analytic properties of the integral were studied, and some convergence theorems are given such as the monotone convergence theorem, the bounded convergence theorem, and the control convergence theorem so as to provide more methods for the study of Denneberg integral.
In order to make the non-additive measure and integral theory more widely applied, the real analysis method is used to transform the decreasing function with negative value as non-negative function so as to prove that the integral of monotone decreasing function on the interval introduced by Denneberg with distribution function is equivalent to the Lebesgue integral. Analytic properties of the integral were studied, and some convergence theorems are given such as the monotone convergence theorem, the bounded convergence theorem, and the control convergence theorem so as to provide more methods for the study of Denneberg integral.
引文
[1] 樊太和, 贺平安. 实变函数论[M]. 北京:清华大学出版社, 2016: 53-133.
[2] Denneberg D. Non-Additive Measure and Integral[M]. Dordrecht: Kluwer Academic Publishers, 1994: 1-59.
[3] Choquet G. Theory of capacities[J]. Annales de l′Institut Fourier, 1953, 5: 131-295.
[4] Wang Z, Klir G J. Generalized Measure Theory[M]. New York: Springer Verlag, 2008: 167-245.
[5] Dubois D, Prade H. Towards fuzzy differential calculus, Part 1: Integration of fuzzy mapping[J]. Fuzzy Sets and Systems, 1982, 8(1): 1-17.
[6] Murofushi T, Sugeno M. An interpretation of fuzzy measure and the Choquet integral as an integral with respect to a fuzzy measure[J]. Fuzzy Sets and Systems, 1989, 29(2): 201-227.
[7] 巩增泰, 郭元伟.区间值和模糊数值函数的Choquet积分[J]. 兰州大学学报, 2009, 45(4): 112-117.
[8] 巩增泰, 寇旭阳.集值函数关于模糊测度Choquet积分的表示和积分原函数性质[J]. 山东大学学报, 2017, 52(8): 1-9.
[9] Klement E P, Mesiar R, Pap E. A universal integral as common frame for Choquet and Sugeno integral[J]. IEEE Transactions on Fuzzy Systems, 2010, 18(1): 178-187.
[10] 徐鹤萍, 张承坤, 李军. 单调测度空间上一般实值可测函数的泛积分[J]. 中国传媒大学学报, 2018, 25(3): 48-54.
[11] Mesia R, Li J, Pap E. Superdecomposition integrals[J]. Fuzzy Sets and Systems, 2015, 259(5): 3-11.
[12] Li J, Mesia R, Klement E P, Pap E. Convergence theorems for monotone measures[J]. Fuzzy Sets and Systems, 2015, 281(C): 103-127.
[13] Even Y, Lehrer E. Decomposition-integral: unifying Choquet and the concave integrals[J]. Economic Theory, 2014, 56(1): 33-58.
[14] Klement E P, Li J, R Mesiar, Pap E. Integrals based on monotone set functions[J]. Fuzzy Sets and Systems, 2015, 281(C): 88-102.
[15] Torra V, Narukawa Y. Numerical integration for the Choquet integral[J]. Information Fusion, 2016, 31(C): 137-145.
[2] Denneberg D. Non-Additive Measure and Integral[M]. Dordrecht: Kluwer Academic Publishers, 1994: 1-59.
[3] Choquet G. Theory of capacities[J]. Annales de l′Institut Fourier, 1953, 5: 131-295.
[4] Wang Z, Klir G J. Generalized Measure Theory[M]. New York: Springer Verlag, 2008: 167-245.
[5] Dubois D, Prade H. Towards fuzzy differential calculus, Part 1: Integration of fuzzy mapping[J]. Fuzzy Sets and Systems, 1982, 8(1): 1-17.
[6] Murofushi T, Sugeno M. An interpretation of fuzzy measure and the Choquet integral as an integral with respect to a fuzzy measure[J]. Fuzzy Sets and Systems, 1989, 29(2): 201-227.
[7] 巩增泰, 郭元伟.区间值和模糊数值函数的Choquet积分[J]. 兰州大学学报, 2009, 45(4): 112-117.
[8] 巩增泰, 寇旭阳.集值函数关于模糊测度Choquet积分的表示和积分原函数性质[J]. 山东大学学报, 2017, 52(8): 1-9.
[9] Klement E P, Mesiar R, Pap E. A universal integral as common frame for Choquet and Sugeno integral[J]. IEEE Transactions on Fuzzy Systems, 2010, 18(1): 178-187.
[10] 徐鹤萍, 张承坤, 李军. 单调测度空间上一般实值可测函数的泛积分[J]. 中国传媒大学学报, 2018, 25(3): 48-54.
[11] Mesia R, Li J, Pap E. Superdecomposition integrals[J]. Fuzzy Sets and Systems, 2015, 259(5): 3-11.
[12] Li J, Mesia R, Klement E P, Pap E. Convergence theorems for monotone measures[J]. Fuzzy Sets and Systems, 2015, 281(C): 103-127.
[13] Even Y, Lehrer E. Decomposition-integral: unifying Choquet and the concave integrals[J]. Economic Theory, 2014, 56(1): 33-58.
[14] Klement E P, Li J, R Mesiar, Pap E. Integrals based on monotone set functions[J]. Fuzzy Sets and Systems, 2015, 281(C): 88-102.
[15] Torra V, Narukawa Y. Numerical integration for the Choquet integral[J]. Information Fusion, 2016, 31(C): 137-145.