What does it mean for a random variable to exhibit zero risk? It of course depends on the risk measure we are using to quantify it.

Let be a risk measure so that

- is a nonatomic probability space
- is law-invariant, i.e., if are two random variables with the same probability distribution, then
- the risk measure is coherent (i.e., it is convex, monotonous, translation invariant and positive homogeneous)
- it is also lower semicontinuous

Take a nonnegative random variable (a.s.). Then if and only if (a.s.).

The proof is quite straightforward and hinges on the fact that under the prescribed assumptions

for all . Assume that . Then, which implies that , thus is almost everywhere zero.

Conversely, if , we may take the random variable which is everywhere zero ( for all ). Then and have the same distribution, so .

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