In lecture notes and the book "Optimal Transport: old and new", the Kantorovich-Rubinstein Duality is presented as:
(K-R) Let $X=Y$ Polish, with $c:X \times X \to \overline{\mathbb R}$ lower-semicontinuous, $\mu, \nu \in \mathcal P_1(X)$. Then $$ \min_{\gamma \in \Pi(\mu,\nu)}\int_{X \times X} c(x,y)d\gamma = \sup\left\{ \int_X f d(\mu -\nu) : ||f||_{Lip_1} \leq 1 \right\} $$
Now, in the book Santambrogio, he does not prove the Kantorovich-Rubinstein Duality per-se, but he gives two theorems that together give something similar to the Kantorovich-Rubinstein.
Thm. 1.42: If $X, Y$ are Polish and $c:X \times Y \to \overline{\mathbb R}$ is l.s.c and bounded from below, then the duality formula $\min KP = \sup DP$.
Prop. 3.1: If $c: X \times X \to \mathbb R$ is a distance, then a function $u: X \to\mathbb R$ is $c$-concave if and only if it is Lipschitz continuous with constant less or equal to 1 with respect to distance $c$.
As is see it, this two results imply the Kantorovich-Rubinstein Duality, but wihtout the need of $\mu,\nu \in \mathcal P_1(X)$, in other words, there is no need for $\int_X |x| d\mu < +\infty$ and $\int_X |y| d\mu < +\infty$.
Now, is my interpretation correct? Why is the $\mu,\nu \in \mathcal P_1(X)$ necessary instead of just $\mu,\nu \in \mathcal P(X)$. What am I missing?