I am interested in two results from optimal transport theory. In particular, I am looking for references showing that the two claims below hold. Let me clarify: I know several books on optimal transport but I am not able to find a reference that clearly shows those claims (probably because obvious - but not to me). Any suggestion (book and chapter) would be extremely appreciated.
Let $\mathcal{I}\equiv \{0,1,...,I\}$, $\mathcal{J}\equiv \{0,1,...,J\}$, $\mathcal{M}\equiv \mathcal{I}\times \mathcal{J}\setminus\{0,0\}$, $\Phi: \mathcal{M}\rightarrow \mathbb{R}$ with $\Phi_{ij}\equiv \Phi(i,j)$ $\forall (i,j)\in \mathcal{M}$.
Consider the following LP problem \begin{equation} \begin{aligned} (1) \hspace{1cm}&\max_{\mu} \sum_{(i,j)\in \mathcal{M}}\mu_{ij}\Phi_{ij}\\ &\text{s.t.}\\ &\mu_{ij}\in \{0,1\} \text{ } \forall (i,j)\in \mathcal{M}\\ &\sum_{j\in \mathcal{J}}\mu_{ij}=1 \text{ } \forall i \in \mathcal{I}\setminus\{0\}\\ &\sum_{i\in \mathcal{I}}\mu_{ij}=1 \text{ } \forall j \in \mathcal{J}\setminus\{0\}\\ \end{aligned} \end{equation} Claim I: If $I< \infty$, $J< \infty$, then (1) has one solution. If $I\rightarrow \infty$ and $J\rightarrow \infty$, then (1) has one solution.
Consider now the dual of (1) \begin{equation} \label{eq2} \begin{aligned} &(2) \hspace{1cm}\min_{U, V} \sum_{i=1}^I U_i+\sum_{j=1}^JV_j\\ &\text{s.t.}\\ &U_i+V_j\geq \Phi_{ij} \text{ }\forall i \in \mathcal{I}, \forall j \in \mathcal{J}\\ & U_i\geq \Phi_{i0}\text{ }\forall i \in \mathcal{I}\\ & V_j\geq \Phi_{0j}\text{ }\forall j \in \mathcal{J} \end{aligned} \end{equation}
Claim II: If $I< \infty$, $J< \infty$, then (2) has at least one solution (product of intervals). If $I\rightarrow \infty$ and $J\rightarrow \infty$, then (2) has one solution.