Proof of A Coupling on Euclidean Space with an Absolutely Continuous First Marginal and Cyclically Monotone Support is Induced by a Map

Each result cited is universally quantified over the data in its own statement, and is applied below with the data named at each use. Write $j:\mathbb{N}\to\mathbb{R}$ for the canonical map of $\mathbb{R}$ (the letter $\iota$ being reserved for the concatenation map of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pairs). Put $\nu=(\mathrm{pr}_{2})_{\#}\theta$, which belongs to $\mathcal{P}(\mathbb{R}^{d})$ by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward, $\mathrm{pr}_{2}$ being Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §projections. Since $(\mathrm{pr}_{1})_{\#}\theta=\mu$ and $(\mathrm{pr}_{2})_{\#}\theta=\nu$, the measure $\theta$ is a coupling of $\mu$ and $\nu$, that is $\theta\in\Pi(\mu,\nu)$; that definition asks nothing beyond $\theta\in\mathcal{P}(\mathbb{R}^{d+d})$ and these two marginal identities.

0. The projections are continuous. For $z,z'\in\mathbb{R}^{d+d}$ one has $\mathrm{pr}_{1}(z)-\mathrm{pr}_{1}(z')=\mathrm{pr}_{1}(z-z')$, because by Concatenation Identifies a Product of Euclidean Spaces with a Euclidean Space the concatenation map reads off components and differences of points of a Euclidean space are taken componentwise (Difference, Dot Product, and Orthogonality in $\mathbb{R}^n$). Hence $\lVert\mathrm{pr}_{1}(z)-\mathrm{pr}_{1}(z')\rVert\le\lVert z-z'\rVert$ by the norm bound of Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §projections, so by Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ §distance the map $\mathrm{pr}_{1}$ is continuous, the choice $\delta=\varepsilon$ serving at every point.

1. A closed cyclically monotone set of full measure. Let $\Gamma=\operatorname{supp}\theta$, the support of the Borel measure $\theta$ on $(\mathbb{R}^{d+d},d_{E})$; it is cyclically monotone by hypothesis, and by The Support of a Borel Measure is Closed, and Carries Full Measure on a Separable Space §closed it is closed and Borel. The metric space $(\mathbb{R}^{d+d},d_{E})$ is separable by Euclidean Space is a Separable Metric Space §separable, so The Support of a Borel Measure is Closed, and Carries Full Measure on a Separable Space §full gives $\theta(\mathbb{R}^{d+d}\setminus\Gamma)=0$, whence $\theta(\Gamma)=\theta(\mathbb{R}^{d+d})-0=1$ by Basic Properties of a Measure §differences. In particular $\Gamma\ne\emptyset$, since the empty set has measure $0$ by Measure, Measure Space, and Probability Measure.

2. The potential. By Rockafellar's Theorem: a Cyclically Monotone Set Lies in the Subdifferential of a Convex Function, applied to the nonempty cyclically monotone set $\Gamma$ of step 1, there are a nonempty convex set $C\subseteq\mathbb{R}^{d}$ and a function $\psi:C\to\mathbb{R}$, convex on $C$, with $\mathrm{pr}_{1}(z)\in C$ by Rockafellar's Theorem: a Cyclically Monotone Set Lies in the Subdifferential of a Convex Function §domain and $\mathrm{pr}_{2}(z)\in\partial_{C}\psi(\mathrm{pr}_{1}(z))$ by Rockafellar's Theorem: a Cyclically Monotone Set Lies in the Subdifferential of a Convex Function §subgradient, for every $z\in\Gamma$.

3. The interior of the domain has full measure. Put $G=\operatorname{int}C$, open by The Interior is the Largest Open Subset and convex by The Interior of a Convex Set is Convex and Carries Each of Its Borel Subsets up to a Null Set §convex-interior. For $k\in\mathbb{N}$ let $\Gamma_{k}=\Gamma\cap\bar{B}(0,j(k))$ if $0<j(k)$, and $\Gamma_{k}=\emptyset$ otherwise. Each nonempty $\Gamma_{k}$ is closed and bounded, hence compact by Heine-Borel Theorem in $\mathbb{R}^n$, so $\mathrm{pr}_{1}(\Gamma_{k})$ is compact by step 0 and Continuous Image of a Compact Space is Compact, and therefore belongs to $\mathcal{B}(\mathbb{R}^{d})$ by Compact Subsets of a Metric Space are Closed and Borel; for empty $\Gamma_{k}$, $\mathrm{pr}_{1}(\Gamma_{k})=\emptyset\in\mathcal{B}(\mathbb{R}^{d})$. By claim 1 of The Archimedean Property of the Real Numbers every $z\in\mathbb{R}^{d+d}$ satisfies $\lVert z\rVert<j(k)$ for some $k\in\mathbb{N}$, so $\Gamma=\bigcup_{k\in\mathbb{N}}\Gamma_{k}$ and

$$E=\bigcup_{k\in\mathbb{N}}\mathrm{pr}_{1}(\Gamma_{k})$$

is the image $\mathrm{pr}_{1}(\Gamma)$ and belongs to $\mathcal{B}(\mathbb{R}^{d})$. By step 2, $E\subseteq C$; and $\Gamma\subseteq\mathrm{pr}_{1}^{-1}(E)$, so

$$\mu(E)=\theta\bigl(\mathrm{pr}_{1}^{-1}(E)\bigr)\ge\theta(\Gamma)=1 ,$$

using $\mu=(\mathrm{pr}_{1})_{\#}\theta$ and Basic Properties of a Measure §monotone; hence $\mu(E)=1$. By The Interior of a Convex Set is Convex and Carries Each of Its Borel Subsets up to a Null Set §null, $\lambda_{d}(E\setminus G)=0$, and $E\setminus G$ is Borel, so $\mu(E\setminus G)=0$ because $\mu$ is absolutely continuous. Basic Properties of a Measure §differences now gives $\mu(E\cap G)=1$, and Basic Properties of a Measure §monotone gives $\mu(G)=1$.

4. Differentiability off a Borel null set. Let $\phi:G\to\mathbb{R}$ be the restriction of $\psi$ to $G$; it is convex on $G$, the defining inequality of Convex Real-Valued Function on a Convex Subset of $\mathbb{R}^n$ holding for points of $G\subseteq C$. Since $G$ is open, $G$ is itself an open subset of $G$ containing each of its points, so every $x\in G$ is an interior point of $G$; hence A Convex Function is Lipschitz on a Ball around an Interior Point, applied to the convex set $G$ and the convex function $\phi$, gives positive reals $\rho,L$ with $\bar{B}(x,\rho)\subseteq G$ and $|\phi(y)-\phi(y')|\le L\lVert y-y'\rVert$ for $y,y'\in\bar{B}(x,\rho)$. Thus $\phi$ is locally Lipschitz on $G$, and Rademacher's Theorem in $\mathbb{R}^n$ §locally-lipschitz, read with $m=1$, shows that the set of points of $G$ at which $\phi$ is not differentiable is null; let $N_{0}\in\mathcal{B}(\mathbb{R}^{d})$ with $\lambda_{d}(N_{0})=0$ contain it, as that notion of nullity provides. Put $D=G\setminus N_{0}\in\mathcal{B}(\mathbb{R}^{d})$. Then $\phi$ is differentiable at every point of $D$, and $G\cap N_{0}$ is Borel and contained in $N_{0}$, so $\mu(G\cap N_{0})=0$ by absolute continuity and Basic Properties of a Measure §monotone; since $D=G\setminus(G\cap N_{0})$, Basic Properties of a Measure §differences gives $\mu(D)=\mu(G)=1$.

5. The map. Let $x\in D$. Since $G$ is open and convex and $\phi$ is convex on $G$ and differentiable at $x$, Elementary Calculus of the Subdifferential of a Convex Function §gradient gives a point $T_{0}(x)\in\mathbb{R}^{d}$, read off from the derivative matrix of $\phi$ at $x$ as described there, with $\partial_{G}\phi(x)=\{T_{0}(x)\}$. This defines $T_{0}:D\to\mathbb{R}^{d}$.

$T_{0}$ is continuous on $D$: given $x\in D$ and a positive real $\varepsilon$, Elementary Calculus of the Subdifferential of a Convex Function §continuity supplies a positive $\delta$ such that every $x'\in G$ with $\lVert x'-x\rVert<\delta$ and every $q\in\partial_{G}\phi(x')$ satisfy $\lVert q-T_{0}(x)\rVert<\varepsilon$; taking $x'\in D$ and $q=T_{0}(x')$ gives the requirement of Continuous Map Between Metric Spaces. By The Metric Subspace: Continuity, the Borel Sigma-Algebra, and the Restriction of a Borel Measure §continuity-map the map $T_{0}$ is then continuous from the metric space $(D,d_{E})$ to $(\mathbb{R}^{d},d_{E})$, hence measurable with respect to $\mathcal{B}(D)$ and $\mathcal{B}(\mathbb{R}^{d})$ by claim 3 of Borel Measurability and Bounded Integration on a Metric Space; and $\mathcal{B}(D)=\{B\in\mathcal{B}(\mathbb{R}^{d}):B\subseteq D\}$ by The Metric Subspace: Continuity, the Borel Sigma-Algebra, and the Restriction of a Borel Measure §borel-subset, $D$ being Borel.

Let $S:\mathbb{R}^{d}\to\mathbb{R}^{d}$ agree with $T_{0}$ on $D$ and take the value $0_{\mathbb{R}^{d}}$ on $\mathbb{R}^{d}\setminus D$. For $B\in\mathcal{B}(\mathbb{R}^{d})$ the preimage $S^{-1}(B)$ is $T_{0}^{-1}(B)$ if $0_{\mathbb{R}^{d}}\notin B$ and $T_{0}^{-1}(B)\cup(\mathbb{R}^{d}\setminus D)$ otherwise; in both cases it belongs to $\mathcal{B}(\mathbb{R}^{d})$. So $S$ is Borel.

6. The measure $\theta$ is concentrated on the graph of $S$. Let $\Gamma_{S}$ be the graph of $S$ in the sense of A Coupling Concentrated on the Graph of a Borel Map is the Push-Forward by That Map, a member of $\mathcal{B}(\mathbb{R}^{d+d})$ as shown there. Put $\Gamma'=\Gamma\cap\mathrm{pr}_{1}^{-1}(D)$, Borel since $\mathrm{pr}_{1}$ is Borel. Since $\theta(\Gamma)=1$ and $\theta(\mathrm{pr}_{1}^{-1}(D))=\mu(D)=1$, and $\theta$ is a probability measure, Basic Properties of a Measure §differences gives $\theta(\mathbb{R}^{d+d}\setminus\Gamma)=0$ and $\theta\bigl(\mathrm{pr}_{1}^{-1}(\mathbb{R}^{d}\setminus D)\bigr)=\theta\bigl(\mathbb{R}^{d+d}\setminus\mathrm{pr}_{1}^{-1}(D)\bigr)=0$. The complement $\mathbb{R}^{d+d}\setminus\Gamma'$ is the union $(\mathbb{R}^{d+d}\setminus\Gamma)\cup\mathrm{pr}_{1}^{-1}(\mathbb{R}^{d}\setminus D)$ of these two Borel sets, so it has measure $0$ by Basic Properties of a Measure §subadditivity, applied to the sequence consisting of these two sets followed by empty sets (each of measure $0$ by Measure, Measure Space, and Probability Measure); a further application of Basic Properties of a Measure §differences gives $\theta(\Gamma')=1$. Let $z\in\Gamma'$ and $x=\mathrm{pr}_{1}(z)\in D\subseteq G\subseteq C$. By step 2, $\mathrm{pr}_{2}(z)\in\partial_{C}\psi(x)$, that is $\psi(u)\ge\psi(x)+\mathrm{pr}_{2}(z)\cdot(u-x)$ for every $u\in C$; restricting $u$ to $G$ gives $\mathrm{pr}_{2}(z)\in\partial_{G}\phi(x)=\{T_{0}(x)\}$ by Subdifferential of a Real-Valued Function on a Convex Subset of $\mathbb{R}^n$ §subdifferential and step 5, and $T_{0}(x)=S(x)$ as $x\in D$. Hence $\mathrm{pr}_{2}(z)=S(\mathrm{pr}_{1}(z))$, so $\Gamma'\subseteq\Gamma_{S}$, and $\theta(\Gamma_{S})=1$ by Basic Properties of a Measure §monotone together with $\theta(\mathbb{R}^{d+d})=1$.

7. Conclusion. The measures $\mu,\nu\in\mathcal{P}(\mathbb{R}^{d})$, the Borel map $S$ and the coupling $\theta\in\Pi(\mu,\nu)$ with $\theta(\Gamma_{S})=1$ satisfy the hypotheses of A Coupling Concentrated on the Graph of a Borel Map is the Push-Forward by That Map, whose clause A Coupling Concentrated on the Graph of a Borel Map is the Push-Forward by That Map §graph gives

$$\theta=(\mathrm{id},S)_{\#}\mu ,$$

which is the assertion.