  Some classical mass transportation problems are investigated in a finitely
additive setting. Let $\Omega=\prod_{i=1}^n\Omega_i$ and
$\mathcal{A}=\otimes_{i=1}^n\mathcal{A}_i$, where
$(\Omega_i,\mathcal{A}_i,\mu_i)$ is a ($\sigma$-additive) probability space for
$i=1,\ldots,n$. Let $c:\Omega\rightarrow [0,\infty]$ be an
$\mathcal{A}$-measurable cost function. Let $M$ be the collection of finitely
additive probabilities on $\mathcal{A}$ with marginals $\mu_1,\ldots,\mu_n$. If
couplings are meant as elements of $M$, most classical results of mass
transportation theory, including duality and attainability of the Kantorovich
inf, are valid without any further assumptions. Special attention is devoted to
martingale transport. Let
$(\Omega_i,\mathcal{A}_i)=(\mathbb{R},\mathcal{B}(\mathbb{R}))$ for all $i$ and
$$M_1=\bigl\{P\in M:P\ll P^*\text{ and }(\pi_1,\ldots,\pi_n)\text{ is a
}P\text{-martingale}\}$$ where $P^*$ is a reference probability on
$\mathcal{A}$. If $M_1\ne\emptyset$, then $$\int c\,dP=\inf_{Q\in M_1}\int
c\,dQ\quad\quad\text{for some }P\in M_1.$$ Conditions for $M_1\ne\emptyset$ are
given as well.
