Theorem 7 — strong duality $R_{\varepsilon,p}(\hat P_N,\ell) = \inf_{\gamma\ge0} \mathbb{E}_{\hat P_N}[\ell_\gamma] + \gamma\varepsilon^p$ on Borel $\mathbb{R}^m$, usc losses, $\varepsilon > 0$

Let $E = \mathbb{R}^m$ with an arbitrary norm (a finite-dimensional real normed space) and its Borel $\sigma$-algebra, let $\Xi \subseteq E$ be closed, let $\hat P_N$ be a probability distribution supported on $\Xi$ ($\hat P_N(\Xi^c) = 0$), let $p \in [1,\infty)$ and $\varepsilon > 0$, and let $\ell : E \to \mathbb{R}$ be a measurable, upper semicontinuous, $\hat P_N$-integrable loss function. Then the worst-case risk over the type-$p$ Wasserstein ball satisfies the strong duality

where $\ell_\gamma$ is the Moreau–Yosida regularization of $\ell$, valued in $(-\infty,+\infty]$ on $\Xi$ (it is $+\infty$ where $\ell$ grows faster than $\|\cdot\|^p$), $\mathbb{E}_{\hat P_N}[\ell_\gamma] \in (-\infty,+\infty]$ is its extended expectation (finite negative part, since $\ell_\gamma \ge \ell$ on $\Xi$), and the identity is read in $[-\infty,\infty]$: both sides may be $+\infty$ simultaneously. The right-hand side is the Lagrangian dual of the worst-case risk evaluation problem, $\gamma$ being the multiplier of the Wasserstein constraint.

Formalization Note. The retired version was false because its $\sigma$-algebra on $E$ was a free parameter: with the trivial $\sigma$-algebra no non-constant loss is integrable under any probability measure, so the integrability-guarded worst-case risk was $-\infty$ while the dual side was $\ge -\|\ell\|_\infty$. The new statement does the following differently. (i) $E$ is a finite-dimensional real normed space with its Borel $\sigma$-algebra ([FiniteDimensional ℝ E] [BorelSpace E]). (ii) The loss class is the paper's: an arbitrary measurable, upper semicontinuous, $\hat P_N$-integrable $\ell$ (p. 1 and Assumption 1, p. 9), instead of the retired narrowing to bounded continuous functions; accordingly $\ell_\gamma$ is extended-real-valued (moreauYosida v2) and $\mathbb{E}_{\hat P_N}[\ell_\gamma]$ is the extended expectation with the paper's p. 2 convention (erealExpectation). (iii) The worst-case risk is the paper's $\sup_{Q \in \mathcal{B}_{\varepsilon,p}} \mathbb{E}_Q[\ell]$ with that same convention (worstCaseRisk/nominalRisk v2) instead of a supremum discarding non-integrable $Q$. (iv) Correction to the printed source: the radius is required to be strictly positive. The paper's framework allows $\varepsilon \ge 0$ (p. 6), but at $\varepsilon = 0$ the printed identity fails: for $p = 1$, $\Xi = \mathbb{R}$, $\hat P_N = \delta_0$ and $\ell(\xi) = \xi^2$ (convex, continuous, $\delta_0$-integrable), the ball $\mathcal{B}_{0,1}(\delta_0)$ is $\{\delta_0\}$ because $W_1$ is a metric (p. 3), so the left-hand side is $0$, whereas $\ell_\gamma(0) = \sup_z z^2 - \gamma|z| = +\infty$ for every $\gamma \ge 0$ and the right-hand side is $+\infty$. Strong duality for every upper semicontinuous $\hat P_N$-integrable loss is the standard result for $\varepsilon > 0$ (Gao–Kleywegt, Math. Oper. Res. 2023, Thm. 1; Blanchet–Murthy, Math. Oper. Res. 2019, Thm. 1), which is what is stated here; the paper's own remark that (10) is solvable "for any $\varepsilon > 0$" shows the same reading. Standing assumptions made explicit: $\Xi$ closed (p. 6); $\hat P_N \in \mathcal{P}(\Xi)$ (the ambiguity set is a ball in $\mathcal{P}(\Xi)$ centred at $\hat P_N$, p. 6; it also forces $\Xi \ne \emptyset$); $\hat P_N$ a probability distribution; $p \in [1,\infty)$; the loss measurable (p. 1) and, by Assumption 1 (p. 9), upper semicontinuous and $\hat P_N$-integrable. Losses are taken real-valued, a special case of the paper's extended-real-valued losses that the retired version already made. No growth or boundedness condition on $\ell$ is imposed.