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