 Solution

Use the Lipschitz constant convention $\Vert f{\Vert }_L={\sup }_{x\ne y}|f(x)-f(y)|/d(x,y)$ in the final quantity; it is a seminorm, with constants having value zero. Also interpret the coupling space as $\mathcal{P}(X\times X)$. The printed $\mathcal{P}(X,Y)$ has an undefined $Y$. The third quantity has a duplicated printed label; it is placed in part iii here, with the fourth in part iv.

First prove weak optimal transport duality. For any feasible continuous pair $f,g$ and any coupling of probability distributions $\pi$ with marginals $P,Q$,Taking the supremum over pairs and infimum over couplings gives $m_d(P,Q)\le W(P,Q)$. These quantities are finite: $P\otimes Q$ is a coupling and the continuous cost $d$ is bounded on the compact metric space $X\times X$. The zero pair is feasible, so $0\le m_d\le W\le \mathrm{diam}X$.

The reverse inequality is established in part ii by finite approximation and a proof of the required finite linear programming duality. Parts iii and iv then show that all the dual formulations agree.