A Coupling on Euclidean Space with an Absolutely Continuous First Marginal and Cyclically Monotone Support is Induced by a 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 $\mathrm{pr}_{1},\mathrm{pr}_{2}:\mathbb{R}^{d+d}\to\mathbb{R}^{d}$ be the coordinate projections for the splitting $d+d$ fixed in Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pairs, which are Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §projections, let push-forwards be those of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward, and write $\mathrm{id}$ for the identity map of $\mathbb{R}^{d}$, which is Borel, being continuous. For a Borel map $S:\mathbb{R}^{d}\to\mathbb{R}^{d}$ the pairing $(\mathrm{id},S)$ is Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §pairing, so $(\mathrm{id},S)_{\#}\mu'\in\mathcal{P}(\mathbb{R}^{d+d})$ for every $\mu'\in\mathcal{P}(\mathbb{R}^{d})$. Let $\theta\in\mathcal{P}(\mathbb{R}^{d+d})$, a Borel measure on the metric space $(\mathbb{R}^{d+d},d_{E})$ with $\theta(\mathbb{R}^{d+d})=1$ by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §measures, let $\operatorname{supp}\theta$ be its support, and let $\mu=(\mathrm{pr}_{1})_{\#}\theta\in\mathcal{P}(\mathbb{R}^{d})$. Suppose that $\mu$ is absolutely continuous and that $\operatorname{supp}\theta$ is cyclically monotone.

1. (Induced by a map)¶ There is a Borel map $S:\mathbb{R}^{d}\to\mathbb{R}^{d}$ with $\theta=(\mathrm{id},S)_{\#}\mu$.