Coherent Risk Measures on General Probability Spaces

We extend the definition of coherent risk measures, as introduced by Artzner, Delbaen, Eber and Heath, to general probability spaces and we show how to define such measures on the space of all random variables. We also give examples that relates the theory of coherent risk measures to game theory and to distorted probability measures. The mathematics are based on the characterisation of closed convex sets Pσ of probability measures that satisfy the property that every random variable is integrable for at least one probability measure in the set Pσ.