Let $(X,\mathcal A)$ and $(Y,\mathcal B)$ be measurable spaces, let $\mu \in \mathcal P(X,\mathcal A)$ and $\nu \in \mathcal P(Y,\mathcal B)$, and let $\Pi(\mu,\nu)$ denote the set of probability measures $\gamma$ on $(X \times Y,\mathcal A \otimes \mathcal B)$ whose first marginal is $\mu$ and whose second marginal is $\nu$. If $\gamma_0,\gamma_1 \in \Pi(\mu,\nu)$ and $t \in [0,1]$, then the measure

\begin{align*} \gamma_t := (1-t)\gamma_0 + t\gamma_1 \end{align*}

belongs to $\Pi(\mu,\nu)$.