Are reference measures of law-invariant functionals unique?

Abstract

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.