Proposition 2 (Absolute Mean Spread), pp. 17–18 — Π_z̃𝒫 is exactly the set of Q with Q[fᵀz̃ ≥ θ] = ρ and absolute mean spread ≤ σ

Let $f\in\mathbb R^P$, $\theta\in\mathbb R$, $\sigma\ge0$ and $\rho\in(0,1)$, and consider the instance of the ambiguity set (4) with auxiliary random variables $\tilde u,\tilde v,\tilde w\in\mathbb R$:

Then

In particular every distribution $\mathbb Q^0$ whose absolute mean spread at $\theta$ is at most $\sigma$ and with $\mathbb Q^0[f^\top\tilde z\ge\theta]=\rho$ is a marginal of a member of $\mathcal P$. Hence an ambiguity set defined by absolute mean spread information is the projection of an instance of the paper's standardized ambiguity set, and worst-case expectation constraints over it fall within the paper's framework.

Formalization Note The conditional means are elementary conditional expectations, $\mathbb E_{\mathbb Q}[f^\top\tilde z\mid A]=\mathbb E_{\mathbb Q}[f^\top\tilde z\,\mathbf 1_A]/\mathbb Q[A]$; both events have probability $\rho$ or $1-\rho$, which are positive because $\rho\in(0,1)$, so no division by zero occurs. An outcome is $\omega=(z,(u,v,w))\in\mathbb R^P\times(\mathbb R\times\mathbb R\times\mathbb R)$, with $\tilde u,\tilde v,\tilde w$ the projections uOf, vOf, wOf; the marginal $\Pi_{\tilde z}\mathbb P$ is the push-forward of $\mathbb P$ under the first projection. Members of the right-hand set are required to have $f^\top\tilde z$ integrable, and members of $\mathcal P$ to have $\tilde w$ integrable: these are the expectations the page writes. The page's "$\mathbb Q^0\in\Pi_{\tilde z}\mathcal P$" is the immediate consequence of the set identity, which is what is stated.