Every coherent risk measure admits a scenario representation

RETIRED — THIS STATEMENT IS FALSE AS WRITTEN. Do not attempt to prove it.

The error is the finiteness of the scenario family. This statement asks for a representation indexed by a Fintype, i.e. a maximum over finitely many probability weight vectors. A maximum of finitely many linear functionals is piecewise linear (polyhedral). But the coherent measures on a finite state space are exactly the support functions of closed convex sets of probability vectors, and such a set need not be a polytope.

Counterexample. On three states, let $c$ be the centroid of the simplex and let $P$ be a closed Euclidean ball of small radius $r$ about $c$, contained in the simplex. Put $\rho(X)=\max_{p\in P}\sum_i p_i(-X_i)$. This is coherent: it is a support function, hence sublinear, so subadditive and positively homogeneous; every $p\in P$ is nonnegative, giving monotonicity; and every $p\in P$ sums to one, giving translation invariance. Explicitly $\rho(X)=\langle c,-X\rangle+r\,\lVert(-X)-\overline{(-X)}\mathbf{1}\rVert_2$, whose Euclidean-norm term is not piecewise linear. So $\rho$ is coherent but is not a maximum over any finite family, and no proof of this statement can exist.

What the corrected statement needs. The scenario family must be allowed to be infinite — a closed convex (equivalently compact convex) set of probability vectors — with the maximum taken as a supremum over that set rather than a Finset.sup' over a Fintype. That requires reworking the RepresentedBy definition accordingly; it is not a repair to this statement.

The converse direction is unaffected and is proved: Repr.repr_coherent shows every finitely-represented functional is coherent (finite representability is sufficient, just not necessary). Repr.worstCase_representedBy gives a genuine instance.

Retired by its author after an independent blind read-back identified the finiteness gap.