# Find random variable given joint law

Let $$(\Omega, \mathcal{F}, \mathbb{P})$$ be a probability space (if necessary we can assume to work with the unit interval with the Lebsegue measure) and let $$X: \Omega \to \mathbb{R}$$ be a random variable with law $$\mu$$ (meaning that $$X_\sharp \mathbb{P}=\mu$$). Suppose it is given a probability measure $$\gamma$$ on $$\mathbb{R}^2$$ with first marginal $$\mu$$ (meaning that $$\pi^1_\sharp \gamma = \mu$$, where $$\pi^1(x,y)=x$$ is the projection on the first factor from $$\mathbb{R}^2$$ to $$\mathbb{R}$$). Can we find a random variable $$Y:\Omega \to \mathbb{R}$$ s.t. $$(X,Y)_\sharp \mathbb{P}=\gamma$$?

I think that the problem is equivalent to the following: given the disintegration of $$\gamma$$ w.r.t. $$\mu$$, call it $$\{\mu_x\}_{x \in \mathbb{R}}$$, can we find a random variable $$Y$$ s.t. $$\mathbb{P}(Y \in B \mid X=x) = \mu_x(B)$$ for $$\mu$$-a.e. $$x \in \mathbb{R}$$ and every $$B \in \mathcal{B}(\mathbb{R})$$?

Edit: I add the following comment that may be helpful. It seems (see Bogachev 10.7.7 Corollary) that a sufficient condition is the existence of a random variable $$Z: \Omega \to [0,1]$$ with uniform distribution independent from the given $$X$$. Is it always possible if the probability space is the unit interval with the Lebesgue measure?

Even when $$(\Omega,\mathcal F, \Bbb P)=([0,1],\mathcal B_{[0,1]},\text{d}x)$$, there does not always exist $$Y$$ for which $$(X,Y)_\sharp \Bbb P=\gamma$$.
Lemma: Let $$(\Omega,\mathcal F, \Bbb P)$$ be a probability space, and $$X$$ be a random variable such $$\sigma(X)=\mathcal F$$. For any event $$A\in \mathcal F$$ which is independent of $$X$$, $$P(A)=0$$ or $$1$$.
Proof: Since $$A\in \sigma(X)$$ and $$A$$ is independent of $$\sigma(X)$$, it follows $$A$$ is independent of itself, proving $$P(A)\in \{0,1\}$$.
Finally, let $$X$$ be the random variable on $$([0,1],\mathcal B_{[0,1]},\text{d}x)$$ defined by $$X(\omega)=\omega$$. Obviously, $$\sigma(X)=\mathcal B_{[0,1]}$$. In the case that $$\gamma=\mu\times \nu$$ is a product measure, where $$\nu$$ is not concentrated on a single point, $$(X,Y)_\sharp \Bbb P=\gamma$$ would imply $$Y$$ is a non-constant random variable is independent of $$X$$. This contradicts our lemma.