Proof of Brenier's Theorem in the Noise Norm: Noise-Optimal Couplings out of a Measure with a Density Relative to a Diagonal Gaussian Measure are Induced by a Unique 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.

Conventions. We work in the setting of Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation, with $c$, $\gamma_{c}$, $\mu$ and $\nu$ as in the statement. As in clause 1 of that setting, for $z\in X\times X$ we write $x=\pi_{1}(z)$ and $y=\pi_{2}(z)$; for $u\in X$ and $k\in\mathbb{N}$, $u_{k}=\langle u,e_{k}\rangle$, and for $n\in\mathbb{N}$ and $w\in\mathbb{R}^{n}$ we write $w_{k}$ ($1\le k\le n$) for the $k$-th entry of $w$, so that $p_{n}(u)=(u_{1},\dots,u_{n})$ by Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §coordinates. The metric space $(X,d)$ is separable by Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §space, and $\mathcal{B}(X\times X)=\mathcal{B}(X)\otimes\mathcal{B}(X)$, with $\pi_{1},\pi_{2}$ Borel, by Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §product-sigma. Accordingly Pairings into a Product, the Graph of a Measurable Map, and Couplings Concentrated on a Graph is applied below with $(X,\mathcal{B}(X))$ in place of $(Y,\mathcal{Y})$ and $(X,d)$ in place of $(Z,d_{Z})$; its projections $\mathrm{pr}_{Y},\mathrm{pr}_{Z}$ are then $\pi_{1},\pi_{2}$, for a map $S:X\to X$ its graph is $\Gamma_{S}=\{z\in X\times X:\ y=S(x)\}$, and its image measures $(\mathrm{id},S)_{\#}\mu$, given by $E\mapsto\mu((\mathrm{id},S)^{-1}(E))$ on $\mathcal{B}(X)\otimes\mathcal{B}(X)=\mathcal{B}(X\times X)$, are the push-forwards of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §pushforward. Likewise Measurable Maps into a Hilbert Space with an Orthonormal Basis: Norms, Inner Products and Linear Combinations, Synthesis from Coordinates, Square-Integrability, and Almost-Everywhere Equality is applied with $X$ in place of $E$, the orthonormal basis $(e_{k})_{k\in\mathbb{N}}$ in place of $(f_{k})_{k\in\mathbb{N}}$ and $(X,\mathcal{B}(X))$ in place of $(S,\mathcal{S})$; measurability of maps $X\to X$ there is then Borel measurability, and the coordinate functions of a map $v:X\to X$ are $x\mapsto\langle v(x),e_{k}\rangle$. Real-valued functions on $X$ are called Borel when measurable with respect to $\mathcal{B}(X)$ and $\mathcal{B}(\mathbb{R})$, which is also the Borel $\sigma$-algebra of the metric space $(\mathbb{R},d_{\mathbb{R}})$, $d_{\mathbb{R}}(s,t)=|s-t|$, by claim 2 of Borel Measurability and Bounded Integration on a Metric Space; and Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions is applied on the measurable space $(X,\mathcal{B}(X))$. For $N\in\mathbb{N}$, $S_{N}:X\to\mathbb{R}$ is the function $S_{N}(u)=\sum_{k=1}^{N}a_{k}^{-1}u_{k}^{2}$ of The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability, applied with the noise weights $a$ and the bound $\bar{a}$ of Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §weights; in particular $0<a_{k}\le\bar{a}$ for every $k\in\mathbb{N}$.

Proof of claim 1.

Step 1 (Head maps). Let $\pi$ be a noise-optimal coupling of $\mu$ and $\nu$. By Noise-Optimal Couplings §optimal, $\pi\in\Pi^{a}(\mu,\nu)$, so by Couplings of Finite Noise Cost and Their Noise Cost §couplings and Couplings of Finite Noise Cost and Their Noise Cost §finite it satisfies $\pi(D_{a})=1$ and $\int_{X\times X}c_{a}\,d\pi<\infty$, and by the definition of a coupling $\pi\in\mathcal{P}(X\times X)$ with $(\pi_{1})_{\#}\pi=\mu$ and $(\pi_{2})_{\#}\pi=\nu$. For every $n\in\mathbb{N}$, claim 1 of lem:noise-head-maps-hilbert, applied with $c$, $\mu$, $\nu$, $\pi$ and $n$ (its hypotheses are those of the present statement together with the noise-optimality of $\pi$), shows that the set $A_{n}$ of the Borel maps $T_{n}:X\to\mathbb{R}^{n}$ with $(\pi_{1},p_{n}\circ\pi_{2})_{\#}\pi=(\mathrm{id},T_{n})_{\#}\mu$ is nonempty. Regarding each $A_{n}$ as a subset of the set of all maps from $X$ into $\bigcup_{m\in\mathbb{N}}\mathbb{R}^{m}$, Axiom of Countable Choice gives a sequence $(T_{n})_{n\in\mathbb{N}}$ with $T_{n}\in A_{n}$ for every $n\in\mathbb{N}$; we fix it. For every $n\in\mathbb{N}$, claim 2 of the same lemma, applied with the same data and this $T_{n}$, shows that the set

$$G_{n}=\{z\in X\times X:\ p_{n}(y)=T_{n}(x)\}$$

belongs to $\mathcal{B}(X\times X)$ and that $\pi(G_{n})=1$.

Step 2 (A set of full measure). The set $D_{a}$ belongs to $\mathcal{B}(X\times X)$ by The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §pairs. Put

$$G=D_{a}\cap\bigcap_{n\in\mathbb{N}}G_{n},$$

which belongs to $\mathcal{B}(X\times X)$ because a $\sigma$-algebra is closed under countable intersections (Sigma-Algebra and Measurable Space). Let $N_{1}=(X\times X)\setminus D_{a}$ and $N_{n+1}=(X\times X)\setminus G_{n}$ for $n\in\mathbb{N}$. Since $\pi$ is finite, the rule for differences gives $\pi(N_{1})=1-\pi(D_{a})=0$ and $\pi(N_{n+1})=1-\pi(G_{n})=0$. The complement $(X\times X)\setminus G$ is $\bigcup_{m\in\mathbb{N}}N_{m}$, so countable subadditivity gives $\pi((X\times X)\setminus G)\le\sum_{m\in\mathbb{N}}\pi(N_{m})=0$, every partial sum being $0$; hence $\pi((X\times X)\setminus G)=0$ and, by the rule for differences again, $\pi(G)=1$.

Step 3 (Coordinates of the head maps). For $k\in\mathbb{N}$ define $\tau_{k}:X\to\mathbb{R}$ by $\tau_{k}(x)=\langle p_{k}^{*}(T_{k}(x)),e_{k}\rangle$. By Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §continuity the map $p_{k}^{*}:(\mathbb{R}^{k},d_{E})\to(X,d)$ and the coordinate function $u\mapsto u_{k}$ on $X$ are Borel, and $T_{k}$ is Borel; so $\tau_{k}$ is Borel by two applications of claim 4 of Borel Measurability and Bounded Integration on a Metric Space. Moreover, by the same claim of Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §continuity, $p_{k}(p_{k}^{*}(w))=w$ for $w\in\mathbb{R}^{k}$, and the $k$-th entry of $p_{k}(u)$ is $u_{k}$; taking $u=p_{k}^{*}(T_{k}(x))$ we get

$$\tau_{k}(x)=\bigl(p_{k}(p_{k}^{*}(T_{k}(x)))\bigr)_{k}=\bigl(T_{k}(x)\bigr)_{k}\qquad(x\in X).\tag{1}$$

Put $g_{k}(x)=\tau_{k}(x)-x_{k}$; then $g_{k}$ is Borel by claim 2 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions. For $N\in\mathbb{N}$ put

$$R_{N}(x)=\sum_{k=1}^{N}a_{k}^{-1}g_{k}(x)^{2}\qquad(x\in X),$$

a Borel function by claims 2 and 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, with $0\le R_{N}(x)\le R_{N+1}(x)$, the added term $a_{N+1}^{-1}g_{N+1}(x)^{2}$ being nonnegative. Let

$$B=\bigcup_{M\in\mathbb{N}}\ \bigcap_{N\in\mathbb{N}}\{x\in X:\ R_{N}(x)\le M\},$$

the natural numbers $M$ being read as real numbers. Each set $\{x\in X:R_{N}(x)\le M\}$ is the complement of $\{x\in X:R_{N}(x)>M\}$, which belongs to $\mathcal{B}(X)$ by the criterion of Measure Spaces and the Lebesgue Integral: Standing Notation §measurable (instantiated at the measurable space $(X,\mathcal{B}(X))$); so $B\in\mathcal{B}(X)$ by Sigma-Algebra and Measurable Space. Thus $x\in B$ exactly when there is $M\in\mathbb{N}$ with $R_{N}(x)\le M$ for every $N\in\mathbb{N}$.

Step 4 (The map $T$). Let $x\in B$ and choose $M\in\mathbb{N}$ as just described. For every $N\in\mathbb{N}$, since $0<a_{k}\le\bar{a}$ and $a_{k}^{-1}g_{k}(x)^{2}\ge0$,

$$\sum_{k=1}^{N}g_{k}(x)^{2}=\sum_{k=1}^{N}a_{k}\,a_{k}^{-1}g_{k}(x)^{2}\le\bar{a}\sum_{k=1}^{N}a_{k}^{-1}g_{k}(x)^{2}=\bar{a}\,R_{N}(x)\le\bar{a}\,M,$$

so the series $\sum_{k=1}^{\infty}g_{k}(x)^{2}$ of nonnegative terms converges by the criterion for nonnegative terms. Let $\hat{g}_{k}=\mathbf{1}_{B}\,g_{k}$, which is Borel by claims 1 and 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions; thus $\hat{g}_{k}(x)=g_{k}(x)$ for $x\in B$ and $\hat{g}_{k}(x)=0$ for $x\notin B$. For every $x\in X$ the series $\sum_{k=1}^{\infty}\hat{g}_{k}(x)^{2}$ converges: for $x\in B$ it is the series just treated, and for $x\notin B$ all its terms are $0$. Hence the synthesis claim of lem:measurable-maps-hilbert-valued-2026a, applied to the functions $\hat{g}_{k}$, shows that for every $x\in X$ the series

$$h(x)=\sum_{k=1}^{\infty}\hat{g}_{k}(x)\,e_{k}$$

converges in $X$, that $h:X\to X$ is Borel, and that $\langle h(x),e_{k}\rangle=\hat{g}_{k}(x)$ for all $x\in X$ and $k\in\mathbb{N}$. The identity map of $X$ is continuous, hence Borel by claim 3 of Borel Measurability and Bounded Integration on a Metric Space; so

$$T:X\to X,\qquad T(x)=x+h(x),$$

is Borel by the operations claim of the same lemma.

For every $x\in X$ one has $T(x)-x=h(x)\in X^{a}$. Indeed, for $N\in\mathbb{N}$,

$$S_{N}(h(x))=\sum_{k=1}^{N}a_{k}^{-1}\hat{g}_{k}(x)^{2},$$

which equals $R_{N}(x)\le M$ if $x\in B$ (with $M$ chosen for $x$ as above) and equals $0$ if $x\notin B$; so the sequence $(S_{N}(h(x)))_{N\in\mathbb{N}}$ is bounded above and $h(x)\in X^{a}$ by The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §partial-sums. Moreover, for $x\in B$ and $k\in\mathbb{N}$, by linearity of the inner product,

$$\langle T(x),e_{k}\rangle=x_{k}+\hat{g}_{k}(x)=x_{k}+g_{k}(x)=\tau_{k}(x).\tag{2}$$

Step 5 ($\pi$ is concentrated on the graph of $T$). Let $z\in G$, with $x=\pi_{1}(z)$ and $y=\pi_{2}(z)$. For every $k\in\mathbb{N}$ we have $z\in G_{k}$, that is $p_{k}(y)=T_{k}(x)$, so by (1) $\tau_{k}(x)=(p_{k}(y))_{k}=y_{k}$, and therefore

$$g_{k}(x)=y_{k}-x_{k}=\langle y-x,e_{k}\rangle .$$

Consequently $R_{N}(x)=\sum_{k=1}^{N}a_{k}^{-1}\langle y-x,e_{k}\rangle^{2}=S_{N}(y-x)$ for every $N\in\mathbb{N}$. Since $z\in D_{a}$, $y-x\in X^{a}$, so by The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §partial-sums the sequence $(S_{N}(y-x))_{N\in\mathbb{N}}$ is bounded above with least upper bound $|y-x|_{a}^{2}$. By claim 1 of The Archimedean Property of the Real Numbers there is $M\in\mathbb{N}$ with $|y-x|_{a}^{2}<M$, and then $R_{N}(x)\le M$ for every $N\in\mathbb{N}$; so $x\in B$. By (2), $\langle T(x),e_{k}\rangle=\tau_{k}(x)=y_{k}$, so $\langle T(x)-y,e_{k}\rangle=0$ for every $k\in\mathbb{N}$, and $T(x)-y=0_{X}$ because $(e_{k})_{k\in\mathbb{N}}$ is an orthonormal basis. Thus $y=T(x)$, that is $z\in\Gamma_{T}$; we have shown $G\subseteq\Gamma_{T}$.

Since $T$ is Borel and $(X,d)$ is separable, Pairings into a Product, the Graph of a Measurable Map, and Couplings Concentrated on a Graph §graph-measurable gives $\Gamma_{T}\in\mathcal{B}(X)\otimes\mathcal{B}(X)=\mathcal{B}(X\times X)$, and monotonicity gives $1=\pi(G)\le\pi(\Gamma_{T})\le\pi(X\times X)=1$, so $\pi(\Gamma_{T})=1$.

Step 6 ($T$ is a noise-optimal map inducing $\pi$). The measure $\pi$ is a probability measure on $(X\times X,\mathcal{B}(X)\otimes\mathcal{B}(X))$ with $(\pi_{1})_{\#}\pi=\mu$ and $\pi(\Gamma_{T})=1$, and $T$ is Borel; so Pairings into a Product, the Graph of a Measurable Map, and Couplings Concentrated on a Graph §graph, applied with $\mu$, $T$ in place of $S$, and $\pi$, gives

$$\pi=(\mathrm{id},T)_{\#}\mu,\qquad \nu=(\pi_{2})_{\#}\pi=T_{\#}\mu .$$

The map $(\mathrm{id},T):X\to X\times X$ is Borel by Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §pairing, and the function $c_{a}$ of The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §pairs is Borel and nonnegative with $c_{a}((\mathrm{id},T)(x))=n_{a}(T(x)-x)$ for every $x\in X$, by the definition of $c_{a}$. Claim 2 of Image Measures, Measures with Densities, and Change of Variables, applied to the measure space $(X,\mathcal{B}(X),\mu)$, the map $(\mathrm{id},T)$ and the function $c_{a}$, therefore gives

$$\int_{X}n_{a}\bigl(T(x)-x\bigr)\,\mu(dx)=\int_{X}c_{a}\circ(\mathrm{id},T)\,d\mu=\int_{X\times X}c_{a}\,d\bigl((\mathrm{id},T)_{\#}\mu\bigr)=\int_{X\times X}c_{a}\,d\pi<\infty,$$

the final inequality by Step 1. Finally $(\mathrm{id},T)_{\#}\mu=\pi$ is noise-optimal. Together with Step 4 ($T$ Borel and $T(x)-x\in X^{a}$ for every $x\in X$) and $T_{\#}\mu=\nu$, this shows that $T$ is a noise-optimal map from $\mu$ to $\nu$ in the sense of Noise-Optimal Maps and Uniquely Noise-Mapped Pairs §map, and $\pi=(\mathrm{id},T)_{\#}\mu$. This proves claim 1.

Proof of claim 2.

Step 7 (A reference coupling and its map). The ordered pair $(\mu,\nu)$ is noise-connected by The Noise Wasserstein Distance is a Metric on the Measures Noise-Connected to the Reference Measure: Existence of Noise-Optimal Couplings, Comparison with the Quadratic Wasserstein Distance and Lower Semicontinuity §connected, so The Noise Wasserstein Distance is a Metric on the Measures Noise-Connected to the Reference Measure: Existence of Noise-Optimal Couplings, Comparison with the Quadratic Wasserstein Distance and Lower Semicontinuity §optimal gives a noise-optimal coupling $\pi_{0}\in\Pi^{a}(\mu,\nu)$; we fix it. By claim 1, applied to $\pi_{0}$, we fix a noise-optimal map $T_{0}$ from $\mu$ to $\nu$ with $\pi_{0}=(\mathrm{id},T_{0})_{\#}\mu$. Now let $\pi$ be an arbitrary noise-optimal coupling of $\mu$ and $\nu$; we show $\pi=\pi_{0}$. Write $W=W_{a}(\mu,\nu)$, the noise Wasserstein distance, so that $\int_{X\times X}c_{a}\,d\pi=I^{a}(\pi)=W^{2}$ and $\int_{X\times X}c_{a}\,d\pi_{0}=I^{a}(\pi_{0})=W^{2}$ by Couplings of Finite Noise Cost and Their Noise Cost §cost and Noise-Optimal Couplings §optimal.

Step 8 (An elementary fact about least upper bounds). Let $(p_{m})_{m\in\mathbb{N}}$ and $(q_{m})_{m\in\mathbb{N}}$ be nondecreasing sequences of real numbers, bounded above, with least upper bounds $p$ and $q$. Then $\tfrac12(p+q)$ is the least upper bound of $\{\tfrac12(p_{m}+q_{m}):m\in\mathbb{N}\}$. Indeed, it is an upper bound since $p_{m}\le p$ and $q_{m}\le q$. Given a real $\varepsilon>0$, choose $m_{1},m_{2}\in\mathbb{N}$ with $p_{m_{1}}>p-\varepsilon$ and $q_{m_{2}}>q-\varepsilon$, and let $m$ be the larger of $m_{1},m_{2}$; by monotonicity $p_{m}\ge p_{m_{1}}$ and $q_{m}\ge q_{m_{2}}$, so $\tfrac12(p_{m}+q_{m})>\tfrac12(p+q)-\varepsilon$, and no number below $\tfrac12(p+q)$ is an upper bound. $(\dagger)$

Step 9 (The midpoint coupling is noise-optimal). Define $\bar{\pi}:\mathcal{B}(X\times X)\to[0,\infty]$ by

$$\bar{\pi}(E)=\tfrac12\bigl(\pi(E)+\pi_{0}(E)\bigr)\qquad(E\in\mathcal{B}(X\times X)),$$

a real number in $[0,1]$ since $\pi,\pi_{0}\in\mathcal{P}(X\times X)$. Then $\bar{\pi}(\varnothing)=0$. Let $(E_{m})_{m\in\mathbb{N}}$ be pairwise disjoint members of $\mathcal{B}(X\times X)$ with union $E$. The partial sums $P_{n}=\sum_{m=1}^{n}\pi(E_{m})$ are real and nondecreasing, and $P_{n}=\pi(\bigcup_{m=1}^{n}E_{m})\le1$ by finite additivity and monotonicity; so by Measure, Measure Space, and Probability Measure the countable additivity of $\pi$ says that $\pi(E)$ is the least upper bound of $\{P_{n}:n\in\mathbb{N}\}$. Likewise $\pi_{0}(E)$ is the least upper bound of the partial sums $Q_{n}=\sum_{m=1}^{n}\pi_{0}(E_{m})$. The partial sums of $\sum_{m}\bar{\pi}(E_{m})$ are $\tfrac12(P_{n}+Q_{n})$, real and bounded above by $1$, and by $(\dagger)$ their least upper bound is $\tfrac12(\pi(E)+\pi_{0}(E))=\bar{\pi}(E)$. So $\bar{\pi}$ is a measure on $\mathcal{B}(X\times X)$, that is a Borel measure, with $\bar{\pi}(X\times X)=1$; thus $\bar{\pi}\in\mathcal{P}(X\times X)$. For $A\in\mathcal{B}(X)$, $\bar{\pi}(\pi_{1}^{-1}(A))=\tfrac12(\mu(A)+\mu(A))=\mu(A)$ and $\bar{\pi}(\pi_{2}^{-1}(A))=\tfrac12(\nu(A)+\nu(A))=\nu(A)$, so $\bar{\pi}\in\Pi(\mu,\nu)$ by Couplings of Two Borel Probability Measures on a Hilbert Space and Their Quadratic Cost §coupling; and $\bar{\pi}(D_{a})=\tfrac12(1+1)=1$.

We compute $\int_{X\times X}c_{a}\,d\bar{\pi}$. By the approximation of nonnegative measurable functions by simple functions, applied on $(X\times X,\mathcal{B}(X\times X))$ to the nonnegative Borel function $c_{a}$, there is a sequence $(s_{m})_{m\in\mathbb{N}}$ of nonnegative simple functions with $s_{m}\le s_{m+1}\le c_{a}$ and $c_{a}(z)$ the least upper bound of $\{s_{m}(z):m\in\mathbb{N}\}$ for every $z$; this sequence does not involve the measure. Let $\lambda$ be any of $\pi,\pi_{0},\bar{\pi}$, and let $s=\sum_{i=1}^{r}t_{i}\mathbf{1}_{A_{i}}$ be the standard representation (Simple Function and Its Integral) of a nonnegative simple function $s$, so that each $t_{i}\ge0$ and $A_{i}\in\mathcal{B}(X\times X)$. By claim 1 of Linearity and Monotonicity of the Lebesgue Integral (applied $r-1$ times to sums and $r$ times to nonnegative multiples) and The Integral of an Indicator Function is the Measure of the Set,

$$\int_{X\times X}s\,d\lambda=\sum_{i=1}^{r}t_{i}\,\lambda(A_{i}),$$

a real number. Since $\bar{\pi}(A_{i})=\tfrac12(\pi(A_{i})+\pi_{0}(A_{i}))$, this gives

$$\int_{X\times X}s_{m}\,d\bar{\pi}=\tfrac12\Bigl(\int_{X\times X}s_{m}\,d\pi+\int_{X\times X}s_{m}\,d\pi_{0}\Bigr)\qquad(m\in\mathbb{N}).$$

By the monotonicity part of claim 1 of Linearity and Monotonicity of the Lebesgue Integral, the sequences $(\int s_{m}\,d\pi)_{m}$ and $(\int s_{m}\,d\pi_{0})_{m}$ are nondecreasing and bounded above by $\int c_{a}\,d\pi=W^{2}$ and $\int c_{a}\,d\pi_{0}=W^{2}$, and by Monotone Convergence Theorem their least upper bounds are $\int c_{a}\,d\pi=W^{2}$ and $\int c_{a}\,d\pi_{0}=W^{2}$. By $(\dagger)$ the least upper bound of $\{\int s_{m}\,d\bar{\pi}:m\in\mathbb{N}\}$ is $\tfrac12(W^{2}+W^{2})=W^{2}$, and by Monotone Convergence Theorem applied with $\bar{\pi}$ that least upper bound is $\int c_{a}\,d\bar{\pi}$. Hence

$$\int_{X\times X}c_{a}\,d\bar{\pi}=W^{2}<\infty .$$

Together with $\bar{\pi}(D_{a})=1$ this shows $\bar{\pi}\in\Pi^{a}(\mu,\nu)$ with $I^{a}(\bar{\pi})=W_{a}(\mu,\nu)^{2}$ (Couplings of Finite Noise Cost and Their Noise Cost §finite, Couplings of Finite Noise Cost and Their Noise Cost §cost), so $\bar{\pi}$ is noise-optimal by Noise-Optimal Couplings §optimal.

Step 10 ($\pi=\pi_{0}$). By claim 1, applied to $\bar{\pi}$, there is a noise-optimal map $\bar{T}$ from $\mu$ to $\nu$ with $\bar{\pi}=(\mathrm{id},\bar{T})_{\#}\mu$; we fix it. Its graph $\Gamma_{\bar{T}}$ belongs to $\mathcal{B}(X\times X)$ by Pairings into a Product, the Graph of a Measurable Map, and Couplings Concentrated on a Graph §graph-measurable, and $(\mathrm{id},\bar{T})(x)=(x,\bar{T}(x))\in\Gamma_{\bar{T}}$ for every $x\in X$, so $\bar{\pi}(\Gamma_{\bar{T}})=\mu(X)=1$ and, by the rule for differences, $\bar{\pi}(F)=0$ for $F=(X\times X)\setminus\Gamma_{\bar{T}}$. Now $\pi(F)\le\pi(F)+\pi_{0}(F)=2\bar{\pi}(F)=0$ and likewise $\pi_{0}(F)\le2\bar{\pi}(F)=0$, so $\pi(\Gamma_{\bar{T}})=\pi_{0}(\Gamma_{\bar{T}})=1$ by the rule for differences. Since $(\pi_{1})_{\#}\pi=(\pi_{1})_{\#}\pi_{0}=\mu$, Pairings into a Product, the Graph of a Measurable Map, and Couplings Concentrated on a Graph §graph, applied with $\mu$, $\bar{T}$ in place of $S$, and $\pi$, respectively $\pi_{0}$, gives

$$\pi=(\mathrm{id},\bar{T})_{\#}\mu=\pi_{0}.$$

Step 11 (Conclusion). By Steps 7 and 10, $T_{0}$ is a noise-optimal map from $\mu$ to $\nu$, fixed before $\pi$ was chosen, and every noise-optimal coupling $\pi$ of $\mu$ and $\nu$ equals $\pi_{0}=(\mathrm{id},T_{0})_{\#}\mu$. By Noise-Optimal Maps and Uniquely Noise-Mapped Pairs §uniquely-mapped, the ordered pair $(\mu,\nu)$ is uniquely noise-mapped. This proves claim 2.