A W2 limit in the two-homogeneous case is globally minimizing

Apply [F1] to obtain weak convergence $h^2({\mu }_t)\rightharpoonup h^2({\mu }_{\mathrm{\infty }})$.

The projected-limit theorem [F2] applies to this weak limit. If it denotes the limit by $\nu$, then for every continuous test on the compact sphere both $\nu$ and $\rho :=h^2({\mu }_{\mathrm{\infty }})$ give the limit of the same integrals $\int fd{h}^2({\mu }_t)$. Testing $f\equiv 1$ shows that both have the same total mass $M$. If $M=0$, both finite nonnegative measures are zero. If $M>0$, then $\nu \mathrm{/}M$ and $\rho \mathrm{/}M$ are Borel probability measures with equal integrals against every bounded continuous test, so [F4] gives $\nu =\rho$. Viewed as a measure on ${\mathbb{R}}^d$, this limit globally minimizes $F$ over all finite nonnegative Borel measures, exactly as asserted by the theorem.

By [F3], the feature and regularizer integrals of ${\mu }_{\mathrm{\infty }}$ equal those of its two-homogeneous projection, so $F({\mu }_{\mathrm{\infty }})=F(h^2({\mu }_{\mathrm{\infty }}))$. By step 2.1, this equals the global minimum of $F$ over finite nonnegative Borel measures on ${\mathbb{R}}^d$, and therefore ${\mu }_{\mathrm{\infty }}$ is itself a global minimizer.