Definition of "deterministic coupling" [Villani]

Solution 1:

The function $(Id,T):X\to X\times Y$ is given by $(Id,T)(x)=(Id(x),T(x))=(x,T(x))$. Now let $E\in\mathcal{X}\otimes\mathcal{Y}$. Then $$(Id,T)^{-1}(E)=\{x\in X:(Id,T)(x)\in E\}$$ $$=\{x\in X:(x,T(x))\in E\}.$$ If we can write $E=A\times B$ with $A\in\mathcal{X}$ and $B\in\mathcal{Y}$, we have

$$(Id,T)^{-1}(E)=\{x\in X:(x,T(x))\in A\times B\}$$

$$ =\{x\in X:x\in A\text{ and }T(x)\in B\}$$

$$ =\{x\in X:x\in A\text{ and }x\in T^{-1}(B)\}$$

$$ =A\cap T^{-1}(B). $$

A probability measure defined on $\mathcal{X}\otimes\mathcal{Y}$ is already determined by its values on "measurable rectangles" $A\times B$, so the last expression suffices already to specify this coupling.