Can the joint law $P \circ (X,Y)^{-1}$ of two random variables $X$ and $Y$ be written as $P \circ (X,\phi(X,U))^{-1}$ for $U$ uniform in $[0,1]$?

Question

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!

Answer 1

Lemma. Let $(\Omega,\mathcal{A},\mathbb{P})$ be a probability space, $X$ a real-valued random variable on this space, and $\mathcal{C} \subset \mathcal{A}$ a $\sigma$-field. Let $F$ be the cumulative distribution function of the conditional law $\mathcal{L}(X \mid \mathcal{C})$. Let $\xi$ be a random variable with uniform law on $[0,1]$ and independent of $\mathcal{C}\vee\sigma(X)$. Define $$ U = F^-(X) + \xi\bigl(F(X) - F^-(X)\bigr). $$ Then $U$ is a random variable independent of $\mathcal{C}$, uniformly distributed on $[0, 1]$, and one has $X = G(U)$ where $G$ is the right-continuous inverse function of $F$.

Proposition. Let $(\Omega,\mathcal{A},\mathbb{P})$ be a probability space. Let $\mathcal{C}\subset\mathcal{A}$ be a $\sigma$-field, $V$ be a real-valued random variable, and define $\mathcal{B} = \mathcal{C}\vee\sigma(V)$. Then there exist an embedding $\iota$ from $(\Omega,\mathcal{A},\mathbb{P})$ to a probability space $(\hat{\Omega},\hat{\mathcal{A}},\hat{\mathbb{P}})$ and a uniform random variable $\hat U$ independent of $\hat{\mathcal{C}}$ such that $\hat{\mathcal{B}} \subset \hat{\mathcal{C}} \vee \sigma(\hat U)$.

The proposition is proved with the help of the lemma. The proofs are given in my paper On standardness and cosiness (Lemma 3.24 and Proposition 3.25).

Apply this proposition to $\mathcal{C} = \sigma(X)$ and $V$ real-valued such that $\sigma(V) = \sigma(Y)$, which exists assuming $E_2$ is a standard Borel space.

Tell me if you need more details.