# Realize a coupling in the target space via a measure on the source space

Consider two product measurable spaces $\left(X \times Y,\mathcal{X} \otimes \mathcal{Y}\right)$, $\left(X' \times Y',\mathcal{X'} \otimes \mathcal{Y'}\right)$ with the usual product sigma-algebra, and a measurable function $g$ mapping from $X \times Y$ to $X' \times Y'$ defined by $x' = g_1(x), y' = g_2(y)$. Notice the assumption of no cross-dependence.

If $\mu$ is a probability measure on $\mathcal{X} \otimes \mathcal{Y}$ then $g$ induces a push-forward joint probability measure $\mu' := \mu g^{-1}$ on $\mathcal{X'} \otimes \mathcal{Y'}$ with marginals $\mu'_X$, $\mu'_Y$.

Let $g, \mu'_X, \mu'_Y$ be fixed and define $\Pi\left(\mu'_X,\mu'_Y\right)$ as the class of couplings of $\mu'_X$, $\mu'_Y$ on $\mathcal{X'} \otimes \mathcal{Y'}$, i.e. all joint probability measures with marginals $\mu'_X$, $\mu'_Y$.

Is it true that any of such coupling is obtainable as the push-forward via $g$ of an appropriate probability measure $\nu$, i.e. if $\Gamma \in \Pi\left(\mu'_X,\mu'_Y\right)$ then $\Gamma = \nu g^{-1}$ for some $\nu$?

If it helps, in applications all spaces are Euclidian of varying dimensionality.

Lemma: Let $(X,\mathcal{B})$ and $(Y,\mathcal{C})$ be analytic Borel spaces and let $\pi$ be a measurable map of $X$ onto $Y$. If $\nu$ is any measure on $\mathcal{C}$ there exists a measure $\mu$ on $\mathcal{B}$ such that $\nu(A)=\mu\big(\pi^{-1}(A)\big)$ for all $A\in\mathcal{C}$.
To apply it, note that Euclidean spaces with their Borel $\sigma$-algebra are analytic Borel spaces, and so are their images in Euclidean spaces (with the relative Borel $\sigma$-algebra) under measurable functions. The assumption of no cross-dependence guarantees that the range of $g$ is a rectangle on which all couplings are supported.