# Push-Forward with Multiple Random Variables

Setup
I'm revisiting some notes from optimal transport and came across a result I'm not sure how to prove. It is an elementary result that, given a measure space ($$\Omega, \mathcal{F}, \mathbb{P}$$), a random variable $$X: \Omega \rightarrow \Omega'$$ with law($$X$$) = $$\mu$$ = $$\mathbb{P}\circ X^{-1}$$, and $$f \in L^{1}(\Omega', \mu)$$, that $$\int\limits_{\Omega} f(X) ~d\mathbb{P} = \int\limits_{\Omega'} f(x) ~d\mu$$ by using the push forward measure and definition of Lebesgue integration.

Question
When it comes to coupling, how do I show $$\mathbb{E}[c(X,Y)] = \int c(X,Y) ~d\mathbb{P} = \int c(x,y) ~d\pi$$ where $$\pi$$ is a coupling of $$(\mu_{1},\mu_{2})$$, i.e. has marginals $$\mu_{1}$$ and $$\mu_{2}$$?

Further
This is mainly to understand to following equality $$\inf\limits_{\pi\in\Pi(\mu_{1},\mu_{2})} \int c(x,y) ~d\pi(x,y) = \inf\limits_{\substack{X,Y\\law(X)=\mu_{1}\\law(Y)=\mu_{2}}}\mathbb{E}[c(X,Y)]$$ where $$X$$ and $$Y$$ are random variables and $$\Pi(\mu_{1},\mu_{2}) = \{\pi : \mathcal{F}_{1} \times \mathcal{F}_{2} \rightarrow \mathbb{R}_{0}^{+} \cup\{\infty\} ~~\vert~ \pi ~\text{couples}~ \mu_{1}~ \text{and}~\mu_{2}\}$$.

From “Optimal Transport Old and New”(Villani):

Let $$(\mathcal{X}, \mu )$$ and $$(\mathcal{Y} , \nu)$$ be two probability spaces. Coupling µ and ν means constructing two random variables $$X$$ and $$Y$$ on some probability space $$(Ω, \mathbb{P} )$$, such that law($$X$$)$$=\mu$$, law($$Y$$)$$= \nu$$.

It’s important to notice that the random variables $$X$$ and $$Y$$ are defined on the same probability space, and they need not be independent!

In fact for most cost functions $$(X,Y)$$ will not be an optimal coupling if they are.

Schematically the situation looks like this:

Where we assume without loss of generality that $$\Omega = \mathcal{X}\times\mathcal{Y}$$.

It might be helpful to consider the following discrete example, where we assume the sigma algebra is the power set and $$I=[0,1]$$:

Let’s say the cost matrix $$C$$ is given by $$c_{i,j} = \frac{1}{4} - \pi_{i,j}$$. Then $$\pi$$ is an optimal coupling.

Find an example for an optimal coupling $$(X,Y)$$. What would happen if $$X$$ and $$Y$$ were independent?

While this is not a proof, I hope it helps you to understand why the infima are equivalent.