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.
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Báo cáo toán học: "The Weak Topology on the Space of Probability Capacities in Rd" 9LHWQDP -RXUQDOVietnam Journal of Mathematics 33:3 (2005) 241–251 RI 0$7+(0$7,&6 ‹ 9$67 The Weak Topology on the Space of Probability Capacities in Rd * 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, 2005Abstract. It is shown that the space of probability capacities in Rd equipped withthe weak topology is separable and metrizable, and contains Rd topologically.1. IntroductionNon-additive set functions plays an important role in several areas of appliedsciences, including Artificial Intelligence, Mathematical Economics and Bayesianstatistics. 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 establishedfor Polish spaces, the fundamental study of capacities has focused on Rd , thed−dimensional Euclidean space (e.g., [7, 9]). In this paper we investigate some topological properties of the space of prob-ability capacities equipped with the weak topology. The main result of this papershows that the space of probability capacities equipped with the weak topologyis separable and metrizable, and contains Rd topologically.2. Notation and ConventionWe first recall some definitions and facts used in this paper. Let K(Rd ), F (Rd ),∗ This work was supported by the National Science Council of Vietnam.242 Nguyen Nhuy and Le Xuan SonG (Rd ), B (Rd ) denote the family of all compact sets, closed sets, open sets andBorel sets in Rd , respectively. By a capacity in Rd we mean a set function T : Rd → R+ = [0, +∞)satisfying the following conditions:(i) T (∅) = 0;(ii) T is alternating of infinite order: For any Borel sets Ai , i = 1, 2, . . . , n; n ≥2, we have n (−1)#I +1 T Ai ≤ T Ai , (2.1) i=1 i∈I I ∈I (n)where I (n) = {I ⊂ {1, . . . , n}, I = ∅} and #I denotes the cardinality of I ;(iii) T (A) = sup{T (C ) : C ∈ K(Rd ), C ⊂ A} for any Borel set A ∈ B (Rd );(iv) T (C ) = inf {T (G) : G ∈ G (Rd ), C ⊂ G}, for any compact set C ∈ K(Rd ). A capacity in Rd is, in fact, a generalization of a measure in Rd . Clearly anycapacity is a non-decreasing set function on Borel sets of Rd . By support of a capacity T we mean the smallest closed set S ⊂ Rd such thatT (Rd \S ) = 0. The support of a capacity T is denoted by supp T . We say thatT is a probability capacity in Rd if T has a compact support and T (supp T ) = 1.By C we denote the family of all probability capacities in Rd . ˜ Let T be a capacity in Rd . Then for any measurable function f : Rd → R+and A ∈ B (Rd ), the function fA : R → R defined by fA (t) = T ({x ∈ A : f (x) ≥ t}) for t ∈ R (2.2)is a non-increasing function in t. Therefore we can define the Choquet integral f dT of f with respect to T byA ∞ ∞ T ({x ∈ A : f (x) ≥ t})dt. f dT = fA (t)dt = (2.3) 0 0 A f dT < ∞, we say that f is integrable. In particular for A = Rd we writeIf A f dT = f dT. RdObserve that if f is bounded, then α T ({x ∈ A : f (x) ≥ t}dt, f dT = (2.4) 0 Awhere α = sup{f (x) : x ∈ A}. In the general case if f : Rd → R is a measurable function, we define f + dT − f − dT, f dT = (2.5) A A A + −where f (x) = max{f (x), 0} and f (x) = max{−f (x), 0}.The Weak Topology on the Space of Probability Capacities in Rd 2433. The Weak Topology on the Space of Probabilitiy CapacitiesLet B be a family of sets of the form +B = {U (T ; f1 , . . . , fk ; : T ∈ C , fi ∈ C0 (Rd ), ˜ > 0, i = 1, . . . , k }, 1, . . . , k) i (3.1)whereU (T ; f1 , . . . , fk ...