Gluing lemmas and Skorohod representations

Let $(mathcalX,mathcalE)$, $(mathcalY,mathcalF)$ and $(mathcalZ,mathcalG)$ be measurable spaces. Suppose we are given two probability measures $gamma$ and $ au$, with $gamma$ defined on $(mathcalX imesmathcalY,,mathcalEotimesmathcalF)$ and $ au$ on $(mathcalX imesmathcalZ,,mathcalEotimesmathcalG)$. Conditions for the existence of random variables $X,,Y,,Z$, defined on the same probability space $(Omega,mathcalA,P)$ and satisfying eginequation* (X,Y)simgammaquad extandquad (X,Z)sim au, endequation* 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 citeBPR10 and citeBPR13.