Are reference measures of law-invariant functionals unique?
研究了定义在随机变量上的泛函，如果其值仅依赖于该随机变量在某个参考概率下的分布，那么是否存在多个这样的参考概率。对于货币风险度量等广泛泛函类，证明了除非泛函是常数或仅依赖于本质下确界和上确界，否则参考概率唯一。
A functional defined on random variables f is law invariant with respect to a reference probability if its value only depends on the distribution of its argument f under that measure. In contrast to most of the literature on the topic, we take a concrete functional as given and ask if there can be more than one such reference probability. For wide classes of functionals – including, for instance, monetary risk measures and return risk measures – we demonstrate that this is not the case unless they are (i) constant, or (ii) more generally depend only on the essential infimum and essential supremum of the argument f . Mathematically, the results leverage Lyapunov's Convexity Theorem.