We consider arbitrary subsets L of random variables defined on an arbitrary non-additive probability space $(\mathrm{\Omega },\mathcal{F},\nu )$. A topology τ on L satisfies Condition BU if every open set in this topology which contains $X\in L$ as a member also contains as a subset some $(c,ϵ)$–ball around X, defined as $B_{c,ϵ}\left(X\right)=\{Y\in L|\nu (|X-Y|\ge c)<ϵ\}$. Condition BU is satisfied by any topology of convergence in non-additive measure ν [21], [18] but also by all coarser topologies. Next we consider preference relations that are continuous with respect to any topology τ satisfying Condition BU. For non-atomic ν we prove that any convex and τ-continuous preference relation over the random variables in a given local cone L must satisfy monotonicity in the local cone order. This monotonicity result comes with surprisingly strong decision theoretic implications: (i) The only τ-continuous and convex preference relation defined over all random variables is the indifference relation; (ii) Any τ-continuous and convex preference relation defined over all positive random variables must satisfy payoff-monotonicity; (iii) Any convex and payoff-monotone preference relation defined over all loss random variables must violate τ-continuity.

我们考虑定义在任意非可加概率空间上的随机变量的任意子集L$(\mathrm{\Omega },\mathcal{F},\nu )$. L上的拓扑τ满足条件 BU 如果该拓扑中的每个开集都包含$X\in 升$ 作为成员还包含作为子集的一些 $(C,\epsilon )$–围绕X球，定义为${乙}_{C,\epsilon }\left(X\right)=\{是\in 升|\nu (|X-是|\ge C)<\epsilon \}$. 条件 BU 满足非加性测度ν [21], [18]收敛的任何拓扑，但也满足所有较粗的拓扑。接下来，我们考虑关于满足条件 BU 的任何拓扑τ是连续的偏好关系。对于非原子ν，我们证明对给定局部锥L 中的随机变量的任何凸和τ连续偏好关系必须满足局部锥阶的单调性。这种单调性结果具有惊人的强大决策理论含义：（i）在所有随机变量上定义的唯一τ -连续和凸偏好关系是无差异关系；(ii) 任何τ- 在所有正随机变量上定义的连续和凸偏好关系必须满足支付单调性；(iii) 在所有损失随机变量上定义的任何凸的和收益单调的偏好关系都必须违反τ -连续性。