Let $X$ be a Polish space, $d$ be a lower semicontinuous metric on $X$ and $\mu,\nu$ be Radon probabilities on $X$. Denote by $Lip(\cdot)$ the Lipschitz constant of a function and by $\Pi(\mu,\nu)$ the set of couplings of $\mu$ and $\nu$.
Then, the fact that $$I) \hspace{5mm} \sup_{\text{Lip}(f) \le 1 ,\\ f \text{ bounded} } \int f d(\mu - \nu) = \inf_{\pi\in \Pi(\mu,\nu)} \int d(x,y) d\pi(x,y),$$with the "$f$ bounded" condition being perfunctory whenever i) $|d|_\infty < \infty$ or ii) $\mu$ and $\nu$ having finite first moment w.r.t $d$, is a well known consequence of the general Kantorovich duality, which (in this setting) says that $$II) \hspace{5mm} \sup_{(\varphi,\psi) \in \Phi_d \cap C_b} \int \varphi d\mu + \int \psi d\nu = \inf_{\pi\in \Pi(\mu,\nu)} \int d(x,y) d\pi(x,y),$$where $\Phi_d \cap C_b := \{ (\varphi, \psi) \in C_b(X,\mathbb{R})^2 : \varphi(x) + \phi(\tilde{x}) \le d(x,\tilde{x}), \mu \times \nu \text{-a.e.}(x,\tilde{x}) \in X \times X \}$).
So in order to prove the first equality out of the second, it suffices to show that the two $\sup$'s above coincide.
This is all very well presented at Villani's "Topics in Optimal Transportation".
Ideally, I would like to know how one can prove the initial equality directly, i.e., without having to resort to the general duality. This is because I'm approaching a situation where I want something similar (in spirit) to I) but I don't know/have a general duality to rely upon. So any direct strategy would be very useful.
However, the previous inquiry, if well-posed, seems to be easy to ask but maybe difficult to answer. So I'm working on and emphasizing here the following question (which already gives me valuable insight):
Considering that proving $\le$ in II) uses few lines of argument, can we show $\le$ in I) directly?
Thanks in advance!