= Solution
Use the <Lipschitz constant> convention $\|f\|_L=\sup_{x\ne y}|f(x)-f(y)|/d(x,y)$ 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 $P,Q$,
$$
\int f\,dP+\int g\,dQ
=\int_{X\times X}(f(x)+g(y))\,d\pi(x,y)
\leq\int_{X\times X}d(x,y)\,d\pi(x,y).
$$
Taking the supremum over pairs and infimum over <couplings> gives \b[$m_d(P,Q)\leq W(P,Q)$]. 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 $0\leq m_d\leq W\leq\operatorname{diam}X$.
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.
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