Corollary 11.31.6AA6333BC6B20

Proof. Symmetry and the triangle inequality follow directly from the definition, and $d_{\gamma ,d}\leq 2M_{\theta ,\gamma }$ because both trace measures have mass $M_{\theta ,\gamma }$. Let $\mathcal F$ denote the family of test functions in the supremum. After rescaling, $\mathcal F$ contains a positive multiple of every Lipschitz function, and Lipschitz functions are uniformly dense in $C(X)$. Hence $\mathcal F$ separates finite measures, and together with the injectivity of $\mathcal T_\gamma $ this shows that $d_{\gamma ,d}$ is a metric. The same density, together with the common mass bound, shows that convergence in $d_{\gamma ,d}$ implies weak convergence of the trace measures. Conversely, $\mathcal F$ is uniformly bounded and equicontinuous, so the Arzelà–Ascoli theorem gives a finite uniform $\epsilon $-net in $\mathcal F$ for every $\epsilon >0$. Weak convergence on the finitely many functions of the net, and the bound $2M_{\theta ,\gamma }\epsilon $ for the remaining error, give convergence in $d_{\gamma ,d}$.

Thus $d_{\gamma ,d}$ is the pullback by the homeomorphism of theorem 11.31.2 of a metric for the weak topology. The resulting metric space is compact, hence complete. For two choices of the auxiliary data, both metrics induce the same compact $L^1$ topology, so the identity map is uniformly continuous in both directions. $\square$