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Several objects in the Extremes literature are special instances ofmax-stable random sup-measures. This perspective opens connections to thetheory of random sets and the theory of risk measures and makes it possible toextend corresponding notions and results from the literature with streamlinedproofs. In particular, it clarifies the role of Choquet random sup-measures andtheir stochastic dominance property. Key tools are the LePage representation ofa max-stable random sup-measure and the dual representation of its taildependence functional. Properties such as complete randomness, continuity,separability, coupling, continuous choice, invariance and transformations arealso analysed.

Max-stable random sup-measures with comonotonic tail dependence