Some properties of distortion risk measures

We give a simplified proof of the fact that law invariant convex risk mea sures automatically have Fatou property, which is first shown by Jouini et al. (Adv. Math. Econ. 9:49–71, 2006). After providing a streamlined proof of Kusuoka's rep resentation theorem of law invariant and comonotonically additive coherent risk mea sures, we prove that a coherent distortion risk measures preserves some well-known stochastic orders.