Every represented risk measure is coherent

A worst case over generalized scenarios is always a coherent measure of risk. This is the constructive half of the central theorem of Artzner et al.: representability implies all four axioms. Each axiom traces to a different part of the definition — subadditivity and positive homogeneity to the maximum-of-linear form, translation invariance to the weights summing to one, monotonicity to their nonnegativity.

The converse — that every coherent measure admits such a representation — is the deeper half and is stated separately as an open problem.

As throughout CoherentRisk, the state space is finite and carries no measure of its own; a scenario is a weighting supplied as data, not derived from a probability space.