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

Throughout, each result cited is universally quantified over the data appearing in its own statement and is applied to the data named here. Write $\Pi(\mu,\nu)$ for the set of couplings of $\mu$ and $\nu$, and $\mathrm{pr}_{1},\mathrm{pr}_{2}:\mathbb{R}^{2}\to\mathbb{R}$ for the coordinate projections. For a Borel map $S:\mathbb{R}\to\mathbb{R}$ let $\Gamma_{S}=\{z\in\mathbb{R}^{2}:\mathrm{pr}_{2}(z)=S(\mathrm{pr}_{1}(z))\}$ be its graph, a Borel set by A Coupling Concentrated on the Graph of a Borel Map is the Push-Forward by That Map.

Step 1 (A construction attached to an optimal coupling). Let $\sigma\in\Pi(\mu,\nu)$ be optimal; we construct from it a Borel map $S$ with $\sigma=(\mathrm{id},S)_{\#}\mu$, nondecreasing on a Borel set of full $\mu$-measure and vanishing off it.

The measure $\sigma$ is a Borel measure on $(\mathbb{R}^{2},d_{E})$, so its support $\Gamma=\operatorname{supp}\sigma$ is defined; it is closed by The Support of a Borel Measure is Closed, and Carries Full Measure on a Separable Space (i) and satisfies $\sigma(\Gamma)=1$ by The Support of a Borel Measure is Closed, and Carries Full Measure on a Separable Space (ii), the metric space $\mathbb{R}^{2}$ being separable by Euclidean Space is a Separable Metric Space. By The Support of an Optimal Coupling is Cyclically Monotone (i) the set $\Gamma$ is cyclically monotone, so Cyclically Monotone Subsets of the Doubled Real Line: Two-Point Monotonicity, Ordering of the Sections, and Countability of the Multi-Valued Abscissae applies to it; let $\Gamma_{x}$ denote its sections and $M_{\Gamma}$ the set of abscissae whose section has two distinct elements.

By Cyclically Monotone Subsets of the Doubled Real Line: Two-Point Monotonicity, Ordering of the Sections, and Countability of the Multi-Valued Abscissae (iii) the set $M_{\Gamma}$ is countable, so Countable Sets are Null for an Atomless Measure, One-Point Sets are Lebesgue Null, and an Absolutely Continuous Measure is Atomless (i), applied to the atomless measure $\mu$, gives $M_{\Gamma}\in\mathcal{B}(\mathbb{R})$ and $\mu(M_{\Gamma})=0$. By A Closed Subset of a Euclidean Space is a Countable Union of Compact Sets, and Its Coordinate Projections are Borel (ii), applied with $q=p=1$ to the closed set $\Gamma$, the image $E_{0}=\mathrm{pr}_{1}(\Gamma)$ belongs to $\mathcal{B}(\mathbb{R})$. Put $E=E_{0}\setminus M_{\Gamma}\in\mathcal{B}(\mathbb{R})$.

Since $\Gamma\subseteq\mathrm{pr}_{1}^{-1}(E_{0})$ and $(\mathrm{pr}_{1})_{\#}\sigma=\mu$, claim 2 of Basic Properties of a Measure and Probability Measures on Euclidean Space and Random Vectors: Standing Notation (iv) give $1=\sigma(\Gamma)\le\sigma(\mathrm{pr}_{1}^{-1}(E_{0}))=\mu(E_{0})$; as $\mu(E_{0})\le\mu(\mathbb{R})=1$ by claim 2 of that lemma, $\mu(E_{0})=1$. Since $E_{0}\cap M_{\Gamma}\subseteq M_{\Gamma}$ and $\mu(M_{\Gamma})=0$, claim 2 of that lemma gives $\mu(E_{0}\cap M_{\Gamma})=0$, so claim 3 of Basic Properties of a Measure gives $\mu(E)=\mu(E_{0})-\mu(E_{0}\cap M_{\Gamma})=1$.

Let $x\in E$. Then $x\in\mathrm{pr}_{1}(\Gamma)$, so $\Gamma_{x}$ is nonempty, and $x\notin M_{\Gamma}$, so by Cyclically Monotone Subsets of the Doubled Real Line: Two-Point Monotonicity, Ordering of the Sections, and Countability of the Multi-Valued Abscissae (iii) the section $\Gamma_{x}$ has exactly one element; write $S(x)$ for it. For $x\in\mathbb{R}\setminus E$ put $S(x)=0$.

$S$ is nondecreasing on $E$ and Borel. Let $x,x'\in E$ with $x\le x'$. If $x=x'$ then $S(x)=S(x')$. If $x<x'$ then $S(x)\in\Gamma_{x}$ and $S(x')\in\Gamma_{x'}$, so $S(x)\le S(x')$ by Cyclically Monotone Subsets of the Doubled Real Line: Two-Point Monotonicity, Ordering of the Sections, and Countability of the Multi-Valued Abscissae (ii). Hence A Function Nondecreasing on a Borel Subset of the Real Line and Vanishing Outside It is Borel (i), applied to $E$ and $S$, gives that $S$ is Borel.

$\sigma$ is carried by the graph of $S$. Let $z\in\Gamma\cap\mathrm{pr}_{1}^{-1}(E)$ and put $x=\mathrm{pr}_{1}(z)\in E$. Then $\mathrm{pr}_{2}(z)\in\Gamma_{x}=\{S(x)\}$, so $z\in\Gamma_{S}$; that is, $\Gamma\cap\mathrm{pr}_{1}^{-1}(E)\subseteq\Gamma_{S}$. The complement of $\Gamma\cap\mathrm{pr}_{1}^{-1}(E)$ in $\mathbb{R}^{2}$ is the union of $\mathbb{R}^{2}\setminus\Gamma$ and $\mathrm{pr}_{1}^{-1}(\mathbb{R}\setminus E)$. The first has $\sigma$-measure $1-\sigma(\Gamma)=0$ by claim 3 of Basic Properties of a Measure, and the second has $\sigma$-measure $\mu(\mathbb{R}\setminus E)=1-\mu(E)=0$ by Probability Measures on Euclidean Space and Random Vectors: Standing Notation (iv) and claim 3 of that lemma. Hence their union is $\sigma$-null by claim 4 of Basic Properties of a Measure, and $\sigma(\Gamma\cap\mathrm{pr}_{1}^{-1}(E))=1$ by claim 3 once more. Since $\Gamma\cap\mathrm{pr}_{1}^{-1}(E)\subseteq\Gamma_{S}$, claim 2 of that lemma gives $\sigma(\Gamma_{S})=1$. Therefore A Coupling Concentrated on the Graph of a Borel Map is the Push-Forward by That Map (i) gives

$$\sigma=(\mathrm{id},S)_{\#}\mu,\qquad S_{\#}\mu=\nu .$$

Step 2 (Claim 1). By Existence of an Optimal Coupling of Two Probability Measures with Finite Second Moment there is an optimal $\pi\in\Pi(\mu,\nu)$. Apply Step 1 to it, obtaining $E\in\mathcal{B}(\mathbb{R})$ with $\mu(E)=1$ and a Borel $T$, nondecreasing on $E$ and vanishing off $E$, with $\pi=(\mathrm{id},T)_{\#}\mu$ and $T_{\#}\mu=\nu$. The coupling $(\mathrm{id},T)_{\#}\mu$ is $\pi$, hence optimal, so $T$ is an optimal map from $\mu$ to $\nu$. This is claim 1.

Step 3 (Claim 2). Let $\pi$ and $T$ be as in Step 2 and let $\pi'\in\Pi(\mu,\nu)$ be optimal. By Nonnegative Combinations of Two Finite Measures, and the Average of Two Couplings (ii) the measure $\hat{\pi}$ with $\hat{\pi}(A)=\tfrac{1}{2}\pi(A)+\tfrac{1}{2}\pi'(A)$ is an optimal coupling of $\mu$ and $\nu$, and $\pi(A)\le2\hat{\pi}(A)$ and $\pi'(A)\le2\hat{\pi}(A)$ for every $A\in\mathcal{B}(\mathbb{R}^{2})$.

Apply Step 1 to $\hat{\pi}$, obtaining a Borel $\hat{T}$ with $\hat{\pi}(\Gamma_{\hat{T}})=1$ and $\hat{\pi}=(\mathrm{id},\hat{T})_{\#}\mu$. Then $\hat{\pi}(\mathbb{R}^{2}\setminus\Gamma_{\hat{T}})=0$ by claim 3 of Basic Properties of a Measure, so

$$\pi(\mathbb{R}^{2}\setminus\Gamma_{\hat{T}})\le2\hat{\pi}(\mathbb{R}^{2}\setminus\Gamma_{\hat{T}})=0,$$

and likewise for $\pi'$. Hence $\pi(\Gamma_{\hat{T}})=\pi'(\Gamma_{\hat{T}})=1$ by claim 3 of Basic Properties of a Measure, and A Coupling Concentrated on the Graph of a Borel Map is the Push-Forward by That Map (i) gives $\pi=(\mathrm{id},\hat{T})_{\#}\mu=\pi'$.

Thus every optimal coupling of $\mu$ and $\nu$ equals $\pi=(\mathrm{id},T)_{\#}\mu$, where $T$ is an optimal map from $\mu$ to $\nu$ by Step 2. By Optimal Transport Maps and Uniquely Mapped Pairs of Probability Measures (ii) the ordered pair $(\mu,\nu)$ is uniquely mapped.