most recent 30 from http://mathoverflow.net 2013-05-21T08:23:14Z http://mathoverflow.net/feeds/question/56853 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/56853/deterministic-coupling-of-probability-measures-with-a-constrained-support-conditi Nick B. 2011-02-27T21:56:22Z 2011-02-27T21:56:22Z

<p>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 $1$. </p> <p>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 $\nu$.</p> <p>Note that this is true in dimension $1$ by taking monotone measure preserving map.</p>