Gluing lemmas and Skorohod representations

Let $(\mathcal{X},\mathcal{E})$, $(\mathcal{Y},\mathcal{F})$ and $(\mathcal{Z},\mathcal{G})$ be measurable spaces. Suppose we are given two probability measures $\gamma$ and $\tau$, with $\gamma$ defined on $(\mathcal{X}\times\mathcal{Y},\,\mathcal{E}\otimes\mathcal{F})$ and $\tau$ on $(\mathcal{X}\times\mathcal{Z},\,\mathcal{E}\otimes\mathcal{G})$. Conditions for the existence of random variables $X,\,Y,\,Z$, defined on the same probability space $(\Omega,\mathcal{A},P)$ and satisfying \begin{equation*} (X,Y)\sim\gamma\quad\text{and}\quad (X,Z)\sim\tau, \end{equation*} are given. The probability $P$ may be finitely additive or $\sigma$-additive. As an application, a version of Skorohod representation theorem is proved. Such a version does not require separability of the limit probability law, and answers (in a finitely additive setting) a question raised in \cite{BPR10} and \cite{BPR13}.