In many books, such as Villni's Optimal Tranport: Old and New, the Kantorovich-Rubinstein duality is written with the condition that $\mu,\nu \in \mathcal P_1(X)$. That is: $$ \int_X |x|d\mu < +\infty \quad , \quad \int_X |x|d\nu < +\infty $$
But, in some Lecture Notes that I found (and using Theorems and Propositions from Santambrogio's book), the Kantorovich-Rubinstein theorem is stated for $\mu,\nu \mathcal P(X)$, hence, without the need of the existence of a first moment. My question is then, is such condition indeed necessary?