Theorem 4.1 — the feasibility sets $K_2^\mu$, $K_2^p$, $K_2^s$ coincide under weak covariance

Consider a stochastic program with fixed recourse (2.1): a fixed $\bar m \times \bar n$ recourse matrix $W$ and random data $\xi = (c, q, p, T)$ with law $\mu$, a probability measure. Let $Q(x,\xi) = \min\{q(\xi)y \mid Wy = p(\xi) - T(\xi)x,\ y \ge 0\}$ (with value $+\infty$ when infeasible and $-\infty$ when unbounded), $\mathcal Q(x) = E_\xi\{Q(x,\xi)\}$ the expected recourse computed with the paper's extended integral (in which $(+\infty) + (-\infty) = +\infty$), and $\tilde\Xi$ the support of $\mu$. Consider the three feasibility sets

1. $K_2^\mu = \{x \mid \text{with probability } 1\ \exists y \ge 0,\ Wy = p(\xi) - T(\xi)x\}$,
2. $K_2^p = \{x \mid \forall \xi \in \tilde\Xi\ \exists y \ge 0,\ Wy = p(\xi) - T(\xi)x\}$,
3. $K_2^s = \{x \mid \mathcal Q(x) < +\infty\}$.

Assume that $\xi$ satisfies the weak covariance condition of Definition 2.2 (all $c_j$, $q_j p_i$ and $q_j t_{ik}$ are integrable) and that $W$ has full row rank $\bar m$. Then

The theorem says that the probabilistic, the support-based and the finite-expected-cost notions of second-stage feasibility agree, so the region where the deterministic equivalent program is finite can be described pointwise on the support.

Formalization Note. The paper writes "such as Definition 2.2"; the statement uses Definition 2.2 itself. The full row rank of $W$ is the paper's standing assumption ("Without loss of generality, we can assume that $W$ has full rank", p. 312); the result remains true without it. $K_2^\mu$ is taken in its first form on p. 314 ($\exists y \ge 0$ almost surely). The paper's remark that $Q(x,\cdot)$ is measurable (p. 313) is a provable fact and is not added as a hypothesis.