C_F(E)上概率测度的紧致性

设C_F(E)是紧度量空间E到Banach空间F上的全体连续映射,赋以一致范数。本文讨论了C_F(E)上概率测度族紧致性的条件,推广了[1]在C[0,1]情形所得到的结果。

Tightness of Probability Measures on C_F(E)

In this paper, We study tightness of a family of probability measures on C_F(E), which is a Banach space of all Continuous mappings of a Compct metric space E into a Banach space F with the uniform norm. We have obtained theorems, which are extensions of two well-known theorems (cf.[1]theorem8.2 and 8.3). Let {P_n} ben Sequence of probability measures on C_F(E) and Suppose that each of the Pn is tight, Then We have Theorem 1. The sequence {P_n} is tight iff these two Conditions hold: (ⅰ) η>0, HC_F(E) Such that P_n(H)>1-η/2, n≥1 and every H(t_i)={x(t_i): x∈H}. is relalively compact, where {t_i} is a Countable dense set in E. (ⅱ) ε>0, η>0, δ, 0<δ<1 and a integer no such that P_n.{x: w_x(δ)≥ε}≥η, n≥n_0 . If E is a convex compact set in an inner prodact space, Then we have Theorem 2. If {P_n} satisfyes(ⅰ) in theorem 1 and (ⅱ) ε>0,η>0, δ, 0<δ<1 and no such that for all t, then {P_n} is tight .