Finitely additive mass transportation

Some classical mass transportation problems are investigated in a finitely additive setting. Let $Ω=\prod_{i=1}^nΩ_i$ and $\mathcal{A}=\otimes_{i=1}^n\mathcal{A}_i$, where $(Ω_i,\mathcal{A}_i,μ_i)$ is a ($σ$-additive) probability space for $i=1,\ldots,n$. Let $c:Ω\rightarrow [0,\infty]$ be an $\mathcal{A}$-measurable cost function. Let $M$ be the collection of finitely additive probabilities on $\mathcal{A}$ with marginals $μ_1,\ldots,μ_n$. 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 $(Ω_i,\mathcal{A}_i)=(\mathbb{R},\mathcal{B}(\mathbb{R}))$ for all $i$ and $M_1=\bigl\{P\in M:P\ll P^*\text{ and }(π_1,\ldots,π_n)\text{ is a }P\text{-martingale}\}$ 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.

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