The following question has come up while working on my senior thesis: Assume $\mu$ and $\nu$ are regular probability measures on $\mathbf{R}^n$. We are also given a coupling $\gamma$ of $\mu$ and $\nu$ such that if $(x,y)\in Supp(\gamma)$ then $|x-y| \leq 1$. So basically given random variable $X$ based on $\mu$ and another one $Y$ based on $\nu$ we know $|X-Y|\leq 1$ with probability $1$.

Now additionally assume $\mu$ is absolutely continuous with respect to Lebesgue measure. Show that there is a deterministic coupling of $\mu$ and $\nu$, i.e. a Borel mapping $s:\mathbf{R}^n \rightarrow \mathbf{R}^n$ such that $s$ pushes $\mu$ forward to $\nu$.

Note that this is true in dimension $1$ by taking monotone measure preserving map.