Question:
Let $S_1 $ and $ S_2$ be two metric spaces. On the space $S_1 \times S_2$, consider two probability measures $P^1$ and $P^2$ such that their first marginal concides: namely $P^1[E\times S_2]=P^2[E\times S_2]$ for each measurable $E\subset S_1$. Without assumption of independece, is it possible (if yes, when?) to find a probability space $(\Omega, \mathcal{F}, P)$ and random variables $A:\Omega \rightarrow S_1 $ and $B,C:\Omega \rightarrow S_2 $ such that $P^1=P\cdot (A,B)^{-1}$ and $P^2=P\cdot (A,C)^{-1}$?
Thanks everybody will give an answear or some suggestions.