Theorem 10 — convex loss and $p = 1$: $R_{\varepsilon,1}(\hat P_N,\ell) = R(\hat P_N,\ell) + \varepsilon\,\mathrm{Lip}(\ell)$ on Borel $\mathbb{R}^m$ (mission IV goal)

Let $E = \mathbb{R}^m$ with an arbitrary norm (a finite-dimensional real normed space) and its Borel $\sigma$-algebra, take $\Xi = E$ (no support constraints), let $\varepsilon \ge 0$, and let $\hat P_N = \frac1N\sum_{i=1}^N \delta_{\hat\xi_i}$ be the empirical distribution of $N \ge 1$ training samples. If the loss function $\ell : E \to \mathbb{R}$ is convex (and $\hat P_N$-integrable, which is automatic), then for $p = 1$ the worst-case risk coincides with the Lipschitz-regularized empirical loss,

an identity in $[-\infty,\infty]$: when $\mathrm{Lip}(\ell) = \infty$ and $\varepsilon > 0$ both sides are $+\infty$ (mass can be pushed to infinity at bounded transport cost), and when $\varepsilon = 0$ both sides equal $R(\hat P_N,\ell)$ (the radius-$0$ ball is $\{\hat P_N\}$ and $0 \cdot \infty = 0$).

Formalization Note. The retired version was false because its $\sigma$-algebra on $E$ was a free parameter: with the $\sigma$-algebra generated by $\{0\}$ on $\mathbb{R}$, every $Q$ under which $\ell = |\cdot|$ is integrable is concentrated at $0$, so the guarded worst-case risk was $\le 0$ while the right-hand side was $\ge \varepsilon\,\mathrm{Lip}(\ell) = 1$. The new statement does the following differently. (i) $E$ is a finite-dimensional real normed space (the paper's $\mathbb{R}^m$ with an arbitrary norm) carrying its Borel $\sigma$-algebra ([FiniteDimensional ℝ E] [BorelSpace E]); the retired version quantified over every MeasurableSpace E, including $\sigma$-algebras unrelated to the norm, for which the Wasserstein cost is a lower integral and non-constant losses are integrable under no measure. (ii) The worst-case risk is the paper's $\sup_{Q \in \mathcal{B}_{\varepsilon,p}} \mathbb{E}_Q[\ell]$ with the paper's p. 2 convention for non-integrable losses (worstCaseRisk/nominalRisk v2, valued in $[-\infty,\infty]$) instead of a supremum that silently discarded every $Q$ under which $\ell$ is not integrable. Standing assumptions made explicit: $\Xi$ closed (p. 6); $\hat P_N$ a probability distribution; $p \in [1,\infty)$ and $\varepsilon \ge 0$ (p. 3, 6); the loss is convex on all of $E$ (the theorem's hypothesis), hence continuous, so the standing measurability and upper-semicontinuity conventions (p. 1; Assumption 1, p. 9) hold automatically and are not repeated; $\hat P_N$-integrability (Assumption 1) is carried for fidelity although automatic for the empirical distribution; $\Xi = \mathbb{R}^m$ is the theorem's own assumption. Losses are taken real-valued, a special case of the paper's extended-real-valued losses that the retired version already made. No correction to the printed source. This statement lives in mission IV's own namespace WassersteinDRO.Regularization and mirrors WassersteinDRO.Duality.convex_loss_p1_v2 with this chapter's own (identically defined) v2 definitions.