Set $\pi=\frac{1}{4}(\delta\_{(1,0)}++\delta\_{(1,3)}+\delta\_{(1,1)}+\delta\_{(2,2)})$. Suppose that $\pi\in\Pi(\mu,\nu)$.


How to get the disintegration of $\pi$ with respect to $\mu$?



How to compute the unique disintegration w.r.t. the first coordinate?
By disintegration is meant that $\pi$ can by broken down as
$$
\pi(dx,dy) =\mu(dx)\pi\_{x}(dy),
$$
with $\pi\_x$ a Borel measurable family of sub-probability measures on $X$.


In your example:
$\pi\_{x\_1}={1\over 2}(\delta\_{0}+\delta\_{1})$ if $x\_1=1$; $=\delta\_{2}$ if $x\_1=2$; and $=0$ otherwise.


Then you have $W\_1(\pi\_{x\_1},\nu) = 1/2$ if $x\_1=1$, and $W\_1(\pi\_{x\_1},\nu)=1$ if $x\_1=2$.


