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Báo cáo toán học: "The Weak Topology on the Space of Probability Capacities in Rd"

Nó chỉ ra rằng không gian của khả năng xác suất trong Rd được trang bị với các cấu trúc liên kết yếu là phân chia và metrizable, và chứa Rd topo. | Vietnam Journal of Mathematics 33 3 2005 241-251 V Í e It ini ai m J o mt r im ai I of MATHEMATICS VAST 2005 The Weak Topology on the Space of Probability Capacities in d Nguyen Nhuy1 and Le Xuan Son2 1 Vietnam National University 144 Xuan Thuy Road Hanoi Vietnam 2 Dept. of Math. University of Vinh Vinh City Vietnam Received April 24 2003 Revised April 20 2005 Abstract. It is shown that the space of probability capacities in equipped with the weak topology is separable and metrizable and contains topologically. 1. Introduction Non-additive set functions plays an important role in several areas of applied sciences including Artificial Intelligence Mathematical Economics and Bayesian statistics. A special class of non-additive set functions known as capacities has been intensively studied during the last thirty years see e.g. 3 5-7 9 . Although some interesting results in the theory of capacities has been established for Polish spaces the fundamental study of capacities has focused on the dimensional Euclidean space e.g. 7 9 . In this paper we investigate some topological properties of the space of probability capacities equipped with the weak topology. The main result of this paper shows that the space of probability capacities equipped with the weak topology is separable and metrizable and contains topologically. 2. Notation and Convention We first recall some definitions and facts used in this paper. Let K F This work was supported by the National Science Council of Vietnam. 242 Nguyen Nhuy and Le Xuan Son B denote the family of all compact sets closed sets open sets and Borel sets in respectively. By a capacity in we mean a set function T 0 satisfying the following conditions i 0 ii T is alternating of infinite order For any Borel sets A . 1 2. n 2 we have T A. - 1 1T A. 2.1 - 1 -El -E- where 1. and denotes the cardinality of iii T a sup T c C A .for any Borel set A A iv T c inf T A . . for any compact set C A . A capacity in is in fact a generalization of a .