Corollary 4.5 — $K_2 = K_2^p = K_2^\mu = K_2^s$ with $K_2$ over the support of $(p,T)$

In the setting of Theorem 4.1 (fixed recourse matrix $W$ of size $\bar m \times \bar n$, random data $\xi = (c, q, p, T)$ with law $\mu$), let $\tilde\Xi_{p,T}$ be the support of the marginal distribution of $(p(\xi), T(\xi))$ and, for $\zeta = (p, T)$, let $K_2(\zeta) = \{x \mid p - Tx \in \operatorname{pos} W\}$, where $\operatorname{pos} W = \{Wy \mid y \ge 0\}$. Put

If $\xi$ satisfies the weak covariance condition (Definition 2.2) and $W$ has full row rank, then

with $K_2^p$, $K_2^\mu$, $K_2^s$ as in Theorem 4.1.

The new content over Theorem 4.1 is that the support of the $(p,T)$-marginal may replace the support of $\xi$: feasibility depends only on the right-hand side and technology matrix. From here on the paper writes $K_2$ for this common set.

Formalization Note. $\tilde\Xi_{p,T}$ is the support of the pushforward of $\mu$ under the coordinate projection $\xi \mapsto (p, T)$, which is continuous and hence measurable. Full row rank of $W$ is the paper's standing assumption (p. 312).