A Coupling Concentrated on the Graph of a Borel Map is the Push-Forward by That Map

In the setting of Probability Measures on Euclidean Space and Random Vectors: Standing Notation, let $d\in\mathbb{N}$ satisfy $1\le d$, let $\mu,\nu\in\mathcal{P}(\mathbb{R}^{d})$, and let $\Pi(\mu,\nu)$ be the set of their couplings; the coordinate projections $\mathrm{pr}_{1},\mathrm{pr}_{2}:\mathbb{R}^{d+d}\to\mathbb{R}^{d}$, the pairing of two Borel maps and push-forwards are those of Probability Measures on Euclidean Space and Random Vectors: Standing Notation (v) and Probability Measures on Euclidean Space and Random Vectors: Standing Notation (iv). Write $\mathrm{id}$ for the identity map of $\mathbb{R}^{d}$, which is Borel, being continuous.

Let $S:\mathbb{R}^{d}\to\mathbb{R}^{d}$ be Borel. The pairing $(\mathrm{id},S):\mathbb{R}^{d}\to\mathbb{R}^{d+d}$ is Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (ii), so the push-forward $(\mathrm{id},S)_{\#}\mu$ is a probability measure on $\mathbb{R}^{d+d}$. The graph of $S$ is

$$\Gamma_{S}=\bigl\{z\in\mathbb{R}^{d+d}:\ \mathrm{pr}_{2}(z)=S\bigl(\mathrm{pr}_{1}(z)\bigr)\bigr\} .$$

It belongs to $\mathcal{B}(\mathbb{R}^{d+d})$: the map $g$ on $\mathbb{R}^{d+d}$ with $g(z)=\lVert\mathrm{pr}_{2}(z)-S(\mathrm{pr}_{1}(z))\rVert^{2}$ is Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (iv), applied to the Borel maps $\mathrm{pr}_{2}$ and $S\circ\mathrm{pr}_{1}$, the latter Borel as a composition of Borel maps; the singleton $\{0\}$ belongs to $\mathcal{B}(\mathbb{R})$ by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (v); and $\Gamma_{S}=g^{-1}(\{0\})$, because by claim 2 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square the number $\lVert a-b\rVert$ is the Euclidean distance of $a$ and $b$, which vanishes exactly when $a=b$ since $d_{E}$ is a metric by $\mathbb{R}^n$ with the Euclidean Distance Is a Complete Metric Space with Coordinatewise Convergence, while a product of two real numbers vanishes exactly when one of them does, by claims 1 and 3 of Zero Products and Elementary Identities in a Field.

Let $\pi\in\Pi(\mu,\nu)$ satisfy $\pi(\Gamma_{S})=1$.

1. (The coupling is the push-forward by the map)(i) $\pi=(\mathrm{id},S)_{\#}\mu$ and $S_{\#}\mu=\nu$.