RISK MEASURES FOR NON-INTEGRABLE RANDOM VARIABLES

We show that when a real-valued risk measure is defined on a solid, rearrangement invariant space of random variables, then necessarily it satisfies a weak compactness, also called continuity from below, property, and the space necessarily consists of integrable random variables. As a result we see that a risk measure defined for, say, Cauchy-distributed random variable, must take infinite values for some of the random variables.