Let $X$ and $\bar X$ be two standard Borel spaces, and let $A\subseteq X\times\bar X$ be an analytic subset of the product space. Let $P$ be any probability measure such that $P(A) = 1$, and denote by $p$ and $\bar p$ marginals of $P$ on $X$ and $\bar X$ respectively. Does there exist a universally measurable map $f:X\to \bar X$ such that $\bar p = p\circ f^{-1}$ and such that the graph of $f$ is contained in $A$?



Existence of a map connecting two marginals of a product measure
No. The measures $p, \bar{p}$ could be almost anything; there's nothing in the given conditions that forces $\bar{p}$ to be a possible pushforward of $p$.


As an extreme example, take $X = \{0\}$ to be a singleton, $\bar{X} = [0,1]$, $A = X \times \bar{X}$, $P= \delta\_0 \times m$ where $m$ is Lebesgue measure on $[0,1]$. Then $p = \delta\_0$, $\bar{p} = m$, but for any $f : X \to \bar{X}$, $p \circ f^{-1}$ can only be a point mass at $f(0)$.


