On the Real Line an Atomless Source is Uniquely Mapped, by a Nondecreasing Optimal Map

In the setting of Probability Measures on Euclidean Space and Random Vectors: Standing Notation, in dimension $d=1$. As in Borel Sigma-Algebra on Euclidean Space and the preamble of Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets, the real line $\mathbb{R}$ is identified with the Euclidean space $\mathbb{R}^{1}$, a point of $\mathbb{R}^{1}$ being read as its sole coordinate, so that $\mathcal{B}(\mathbb{R}^{1})=\mathcal{B}(\mathbb{R})$, $\mathcal{P}(\mathbb{R}^{1})=\mathcal{P}(\mathbb{R})$, $\mathcal{P}_{2}(\mathbb{R}^{1})=\mathcal{P}_{2}(\mathbb{R})$ and $\mathbb{R}^{1+1}=\mathbb{R}^{2}$, and the Euclidean distance of $\mathbb{R}^{1}$ is the absolute-value metric by The Euclidean Distance on the Real Line is the Absolute Value Metric.

Let $\mu,\nu$ belong to the set $\mathcal{P}_{2}(\mathbb{R})$ of probability measures with finite second moment and suppose that $\mu$ is atomless.

1. (A nondecreasing optimal map)¶ There are a set $E\in\mathcal{B}(\mathbb{R})$ with $\mu(E)=1$ and a Borel map $T:\mathbb{R}\to\mathbb{R}$ such that $T(x)\le T(x')$ for all $x,x'\in E$ with $x\le x'$, such that $T(x)=0$ for every $x\in\mathbb{R}\setminus E$, and such that $T$ is an optimal map from $\mu$ to $\nu$.

2. (Unique mapping)¶ The ordered pair $(\mu,\nu)$ is uniquely mapped.