On a relationship between distorted and spectral risk measures

We study the relationship between two widely used risk measures, the spectral measures and the distortion risk measures. In both cases, the risk measure can be thought of as a re-weighting of some initial distribution. We prove that spectral risk measures are equivalent to distorted risk pricing measures, or equivalently, spectral risk functions are related to distortion functions. Besides that we prove that distorted measures are absolutely continuous with respect to the original measure. This allows us to find a link between the risk measures based on relative entropy and spectral risk measures or measures based on distortion risk function.