I'm currently reading through "Optimal transport, old and new" by Cédric Villani. In the first chapter, he defines a coupling of two probability spaces $(\mathcal{X},\mu)$ and $(\mathcal{Y},\nu)$ as a probability measure $\pi$ defined on $\mathcal{X} \times \mathcal{Y}$ such that $\pi$ admits $\mu$ and $\nu$ as marginals over $\mathcal{X}$ and $\mathcal{Y}$ respectively.
This is fine, I understand it. However, he then goes on to define a special kind of coupling, a deterministic coupling of $(\mathcal{X},\mu)$ and $(\mathcal{Y},\nu)$ which satisfies:
- There is a measurable function $T$ such that $\mathcal{Y} = T(\mathcal{X})$
- $\pi =(Id,T)_\# \mu $ (where # denotes the pushforward measure).
Point number 2 is what confuses me. $\pi$ is defined over $\mathcal{X} \times \mathcal{Y}$, however for any subset $E$ of $\mathcal{X} \times \mathcal{Y}$, $(Id,T)_\# \mu (E) = \mu((Id,T)^{-1}(E))$ but $\mu$ is only defined over $\mathcal{X}$, so I'm not sure how $\mu((Id,T)^{-1}(E))$ makes sense? Also, I'm unsure how to interpret $(Id,T)^{-1}(E)$.
If anyone is familiar with the above terminology I would greatly appreciate some guidance.
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.