Optimal Transport Maps and Uniquely Mapped Pairs of Probability Measures

In the setting of Probability Measures on Euclidean Space and Random Vectors: Standing Notation, let $d\in\mathbb{N}$ satisfy $1\le d$ and let $\mu,\nu$ belong to the set $\mathcal{P}_{2}(\mathbb{R}^{d})$ of probability measures with finite second moment, with $\Pi(\mu,\nu)$ the set of their couplings. Write $\mathrm{id}$ for the identity map of $\mathbb{R}^{d}$, which is Borel, being continuous. For a Borel map $T:\mathbb{R}^{d}\to\mathbb{R}^{d}$ the pairing $(\mathrm{id},T)$ is Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §pairing, and the push-forward $(\mathrm{id},T)_{\#}\mu$ of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward belongs to $\Pi(\mu,T_{\#}\mu)$ by Couplings on Euclidean Space: Product Coupling, Swap, Finiteness of the Cost, Push-Forward Couplings, Modifying One Marginal, Quantisation, Gluing over a Finitely Supported Measure, and the Lipschitz Bound §pushforward; when $T_{\#}\mu=\nu$ it is therefore a coupling of $\mu$ and $\nu$, of which optimality may be asked.

1. (Optimal map)¶ A Borel map $T:\mathbb{R}^{d}\to\mathbb{R}^{d}$ is an optimal map from $\mu$ to $\nu$ if $T_{\#}\mu=\nu$ and the coupling $(\mathrm{id},T)_{\#}\mu$ is optimal.

2. (Uniquely mapped pair)¶ The ordered pair $(\mu,\nu)$ is uniquely mapped if there is an optimal map $T$ from $\mu$ to $\nu$ such that every optimal coupling of $\mu$ and $\nu$ equals $(\mathrm{id},T)_{\#}\mu$.