This paper considers a decision maker whose preferences are locally upper- or/and lower-semicontinuous in measure. We introduce the notion of a rich set which encompasses any standard vector space of random variables but also much smaller sets containing only random variables with at most two different outcomes in their support. Whenever preferences are complete on a rich set of random variables, lower- (resp. upper-) semicontinuity in measure becomes incompatible with convexity of strictly better (resp. worse) sets. We discuss implications for utility representations and risk-measures. In particular, we show that the value-at-risk criterion violates convexity exactly because it is lower-semicontinuous in measure.

本文考虑的是决策者，其偏爱在度量上局部地处于上半连续或下半连续。我们引入了丰富集合的概念，该集合包含任何随机变量的标准向量空间，也包含仅包含随机变量且支持最多两个不同结果的较小集合。每当对大量随机变量的偏好完成时，度量中较低（分别为较高）的半连续性与严格较好（分别为较差）的凸性不相容。我们讨论了公用事业表示和风险衡量的含义。特别是，我们证明了风险价值准则恰好违反了凸性，因为它在度量上是下半连续的。