Partial one-homogeneous energy is continuous under W2 convergence

First, $\mathrm{\Phi }$ is Borel measurable with values in the separable space $\mathcal{H}$. Fix a countable dense sequence $(h_j)$ in $\mathcal{H}$; the map assigning to $\mathrm{\Phi }(u)$ a nearest point among $h_1,\dots ,h_k$, breaking ties by least index, is finite-valued and measurable, and its error tends to zero as $k\to \mathrm{\infty }$. Thus $\mathrm{\Phi }$ is strongly measurable. Moreover, $\mathrm{\parallel }\mathrm{\Phi }(w,\theta )\mathrm{\parallel }\le \mathrm{\parallel }\phi {\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{\mid }w\mathrm{\mid }$. A probability law with finite second moment has $\int \mathrm{\mid }w\mathrm{\mid }d\mu \le (\int \mathrm{\mid }w{\mathrm{\mid }}^{2}d\mu {)}^{1\mathrm{/}2}<\mathrm{\infty }$ by Cauchy–Schwarz. Thus [F3] gives existence of its feature moment, while $0\le V\le \mathrm{\parallel }\tilde{V}{\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{\mid }w\mathrm{\mid }$ gives a finite penalty; $R$ is finite-valued. This proves the asserted well-definedness for every law in the Statement. By [A1] and [F1] choose couplings ${\pi }_n$ of ${\mu }_n$ and $\mu$ with ${\delta }_n^2=\int (\mathrm{\mid }w-w^{\mathrm{\prime }}{\mathrm{\mid }}^2+\mathrm{\mid }\theta -{\theta }^{\mathrm{\prime }}{\mathrm{\mid }}^2)d{\pi }_n\to 0$. By [F2], the coupling marginal identities hold also for these Bochner feature integrals: apply a bounded linear functional, use the scalar marginal identity, and identify the vectors by all such evaluations. Consequently [F4] and the pointwise bound $\mathrm{\parallel }w\phi (\theta )-w^{\mathrm{\prime }}\phi ({\theta }^{\mathrm{\prime }})\mathrm{\parallel }\le \mathrm{\parallel }\phi {\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{\mid }w-w^{\mathrm{\prime }}\mathrm{\mid }+\mathrm{Lip}(\phi )\mathrm{\mid }{w}^{\mathrm{\prime }}\mathrm{\mid }\mathrm{\mid }\theta -{\theta }^{\mathrm{\prime }}\mathrm{\mid }$ give $\mathrm{\parallel }{m}_{{\mu }_n}-m_{\mu }\mathrm{\parallel }\le (\mathrm{\parallel }\phi {\mathrm{\parallel }}_{\mathrm{\infty }}+\mathrm{Lip}(\phi )(\int \mathrm{\mid }{w}^{\mathrm{\prime }}{\mathrm{\mid }}^{2}d\mu (w^{\mathrm{\prime }},{\theta }^{\mathrm{\prime }}){)}^{1\mathrm{/}2}){\delta }_n⟶0.$ The second-moment factor belongs to the fixed limit marginal and is finite, with no bound on the diameter or interior of $\mathrm{\Theta }$ required.

Since $\mid \mathrm{\mid }w\mathrm{\mid }-\mathrm{\mid }{w}^{\mathrm{\prime }}\mathrm{\mid }\mid \le \mathrm{\mid }w-w^{\mathrm{\prime }}\mathrm{\mid }$, the same estimate with scalar $\tilde{V}$ in place of $\phi$ gives $\int Vd{\mu }_n\to \int Vd\mu$. Step 1.1 gives convergence of feature moments in $\mathcal{H}$; continuity of $R$ therefore gives convergence of the risk terms. Adding the two limits proves $F({\mu }_n)\to F(\mu )$. Under the full feature-energy hypotheses, the identical feature and penalty integrals are finite by step 1.1, so the finite branch in that definition gives exactly this $F$. If $\mathrm{\Theta }=\mathrm{\varnothing }$, no law in the Statement exists and the assertion is vacuous; zero maps, zero weights and singleton or lower-dimensional $\mathrm{\Theta }$ are included in the estimates.