I want to know whether there is some general assumpitons we can make on two measurable spaces $E$ and $F$ (e.g. polish, complete, separable,...) such that we can ensure that the following "Theorem" holds:

Theorem: Suppose that $X: \Omega \rightarrow E$ and $Y: \Omega \rightarrow F$ are two random variables on a probability space $(\Omega, \mathcal{F},P)$. Then there exists a probability space $(\hat{\Omega}, \hat{\mathcal{F}}, \hat{P})$, random variables $\hat{X}:\hat{\Omega} \rightarrow E$ and $\hat{U}: \hat{\Omega} \rightarrow [0,,1]$ with $\hat{P} \circ X^{-1} = P \circ X^{-1}$ and $\hat{U}$ uniformly distributed in $[0,1]$, and a product-measurable mapping $\phi : E \times [0,1] \rightarrow F$ such that $$ P \circ (X,Y)^{-1} = \hat{P} \circ (\hat{X}, \phi(\hat{X}, \hat{U}))^{-1}.$$

I saw something similar when proving existence of discrete time markov chains, where $E$ was assumed to be at most countable and endowed with the discrete $\sigma$-algebra. And I also came across this paper by Skorohod which assumes (in the above setting) that $E=F$ is a Polish space and proves an even stronger result then the above. **But what general assumptions can we make about $E$ and $F$ such that the above can hold?**

Thanks a lot in advance!