  In the present paper we study quantile risk measures and their domain. Our
starting point is that, for a probability measure $ Q $ on the open unit
interval and a wide class $ \mathcal{L}_Q $ of random variables, we define the
quantile risk measure $ \varrho_Q $ as the map which integrates the quantile
function of a random variable in $ \mathcal{L}_Q $ with respect to $ Q $. The
definition of $ \mathcal{L}_Q $ ensures that $ \varrho_Q $ cannot attain the
value $ +\infty $ and cannot be extended beyond $ \mathcal{L}_Q $ without
losing this property. The notion of a quantile risk measure is a natural
generalization of that of a spectral risk measure and provides another view at
the distortion risk measures generated by a distribution function on the unit
interval. In this general setting, we prove several results on quantile or
spectral risk measures and their domain with special consideration of the
expected shortfall.
