Fix a probability space $(\Omega,\mathcal F,\mathbb P)$, and let $X,Y:\Omega\to\mathbb R$ be random variables with laws $\mu_X,\mu_Y$ respectively. Let $\pi$ be a probability measure on $\mathbb R^2$ with marginal distributions $\mu_X,\mu_Y$. Does there exist a random variable $Z:\Omega\to\mathbb R^2$ such that $\mu_Z=\pi$?
Intuitively this seems true, and if it is true, it seems like something that would be a classical well-known fact in probability. However, I have not read any such statement anywhere.