p. 12 — the ambiguity set 𝒫 is contained in every outer approximation 𝒫^j

Let $\mathcal P$ be the standardized ambiguity set (4) and let $\{\mathcal I_j\}_{j\in\mathcal J}$ be a partition of the index set of its confidence sets, given by a block map. Then for every block $j$,

where $\mathcal P^j$ imposes only the probability conditions $\mathbb P[(\tilde z,\tilde u) \in \mathcal C_i] \in [\underline p_i,\overline p_i]$ for $i \in \mathcal I_j$, together with the expectation condition of $\mathcal P$.

This inclusion is what makes the naïve approximation (6) conservative, and it is the first inequality in the proof of Theorem 3.

Formalization Note The statement holds for every block map (surjective or not), without any of the standing conditions (C1)–(C3).