Proof of Pairings of Borel Maps as Couplings, and Gluing Two Couplings over a Finitely Supported Middle Marginal

Each result cited is universally quantified over the data in its own statement, and is applied here to the data named in the statement above.

Claim 1. The projections $\mathrm{pr}^{d,d}_{1}$, $\mathrm{pr}^{d,d}_{2}$ and $\mathrm{pr}^{d+d,d}_{1}$, $\mathrm{pr}^{d+d,d}_{2}$ are Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (i), so each $\mathrm{q}_{i}$ is Borel, a composition of Borel maps being Borel by Probability Measures on Euclidean Space and Random Vectors: Standing Notation (ii); the three pairings are then Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (ii).

Let $x,y,z\in\mathbb{R}^{d}$. By Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (i) applied to the splitting $(d+d)+d$,

$$\mathrm{pr}^{d+d,d}_{1}\bigl(\iota_{3}(x,y,z)\bigr)=\iota^{d,d}(x,y),\qquad\mathrm{pr}^{d+d,d}_{2}\bigl(\iota_{3}(x,y,z)\bigr)=z ,$$

and by the same clause applied to the splitting $d+d$, $\mathrm{pr}^{d,d}_{1}(\iota^{d,d}(x,y))=x$ and $\mathrm{pr}^{d,d}_{2}(\iota^{d,d}(x,y))=y$. Composing gives $\mathrm{q}_{1}(\iota_{3}(x,y,z))=x$, $\mathrm{q}_{2}(\iota_{3}(x,y,z))=y$ and $\mathrm{q}_{3}(\iota_{3}(x,y,z))=z$.

Let $w\in\mathbb{R}^{3d}$. By Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (i) for the splitting $(d+d)+d$, $w=\iota^{d+d,d}(\mathrm{pr}^{d+d,d}_{1}(w),\mathrm{pr}^{d+d,d}_{2}(w))$, and by the same clause for the splitting $d+d$ applied to the point $\mathrm{pr}^{d+d,d}_{1}(w)$ of $\mathbb{R}^{d+d}$,

$$\mathrm{pr}^{d+d,d}_{1}(w)=\iota^{d,d}\bigl(\mathrm{q}_{1}(w),\mathrm{q}_{2}(w)\bigr).$$

Hence $w=\iota_{3}(\mathrm{q}_{1}(w),\mathrm{q}_{2}(w),\mathrm{q}_{3}(w))$, so $\iota_{3}$ is surjective onto $\mathbb{R}^{3d}$; and it is injective, because the preceding paragraph recovers each of $x,y,z$ from $\iota_{3}(x,y,z)$. Thus $\iota_{3}$ is a bijection.

Claim 2. The pairing $(S,T)$ is Borel, as recorded in the statement. For Borel $g:\mathbb{R}^{n+n}\to\mathbb{R}^{n}$ and $B\in\mathcal{B}(\mathbb{R}^{n})$ one has, directly from the definition of a push-forward in Probability Measures on Euclidean Space and Random Vectors: Standing Notation (iv) and the identity $(g\circ(S,T))^{-1}(B)=(S,T)^{-1}(g^{-1}(B))$ for preimages,

$$g_{\#}\bigl((S,T)_{\#}\lambda\bigr)(B)=\bigl((S,T)_{\#}\lambda\bigr)\bigl(g^{-1}(B)\bigr)=\lambda\bigl((g\circ(S,T))^{-1}(B)\bigr)=\bigl(g\circ(S,T)\bigr)_{\#}\lambda(B).$$

Taking $g=\mathrm{pr}^{n,n}_{1}$ and $g=\mathrm{pr}^{n,n}_{2}$, and using $\mathrm{pr}^{n,n}_{1}\circ(S,T)=S$ and $\mathrm{pr}^{n,n}_{2}\circ(S,T)=T$ from Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (i), gives

$$(\mathrm{pr}^{n,n}_{1})_{\#}\bigl((S,T)_{\#}\lambda\bigr)=S_{\#}\lambda,\qquad(\mathrm{pr}^{n,n}_{2})_{\#}\bigl((S,T)_{\#}\lambda\bigr)=T_{\#}\lambda,$$

which is exactly the condition of Couplings of Two Probability Measures on Euclidean Space and Their Quadratic Cost (i) for $(S,T)_{\#}\lambda$ to be a coupling of $S_{\#}\lambda$ and $T_{\#}\lambda$; that it is a probability measure on $\mathbb{R}^{n+n}$ holds by Probability Measures on Euclidean Space and Random Vectors: Standing Notation (iv).

For the cost, Couplings of Two Probability Measures on Euclidean Space and Their Quadratic Cost (ii) gives

$$I\bigl((S,T)_{\#}\lambda\bigr)=\int_{\mathbb{R}^{n+n}}\bigl\lVert\mathrm{pr}^{n,n}_{1}(u)-\mathrm{pr}^{n,n}_{2}(u)\bigr\rVert^{2}\,\bigl((S,T)_{\#}\lambda\bigr)(du),$$

and the change-of-variables formula of Probability Measures on Euclidean Space and Random Vectors: Standing Notation (iv), applied to the Borel map $(S,T)$ and to that nonnegative Borel integrand, turns the right-hand side into $\int_{\mathbb{R}^{m}}\lVert S-T\rVert^{2}\,d\lambda$, since the integrand evaluated at $(S,T)(s)$ is $\lVert S(s)-T(s)\rVert^{2}$ by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (i).

Claim 3. Since $\rho(\mathbb{R}^{d}\setminus F)=0$ and $\rho(\mathbb{R}^{d})=1$, claim 3 of Basic Properties of a Measure gives $\rho(F)=1$; and $F$ being finite with its one-point subsets Borel and pairwise disjoint, claim 1 of Basic Properties of a Measure gives $\rho(F)=\sum_{a\in F}\rho(\{a\})$. Let $F'=\{a\in F:0<\rho(\{a\})\}$, a finite set; the terms of that sum indexed outside $F'$ are $0$, so by the same clause $\rho(F')=1$, and $F'$ is nonempty. For $a\in F'$ write $w_{a}=\rho(\{a\})$, a positive real number.

The conditional measures. Fix $a\in F'$ and set

$$E_{a}=(\mathrm{pr}_{2})^{-1}(\{a\})\subseteq\mathbb{R}^{d+d},\qquad G_{a}=(\mathrm{pr}_{1})^{-1}(\{a\})\subseteq\mathbb{R}^{d+d},$$

Borel sets, being preimages of the Borel set $\{a\}$ under Borel maps. By Couplings of Two Probability Measures on Euclidean Space and Their Quadratic Cost (i), $\pi_{12}(E_{a})=\rho(\{a\})=w_{a}$ and $\pi_{23}(G_{a})=w_{a}$.

By claim 1 of Assembly of Measure Spaces: Restriction, Transport, One-Point Spaces, and Countable Disjoint Unions, applied to the measure space $(\mathbb{R}^{d+d},\mathcal{B}(\mathbb{R}^{d+d}),\pi_{12})$ and the Borel set $E_{a}$, the trace $\sigma$-algebra $\mathcal{B}(\mathbb{R}^{d+d})|_{E_{a}}$ and the restriction $\pi_{12}|_{E_{a}}$ form a measure space on $E_{a}$. The restriction of $\mathrm{pr}_{1}$ to $E_{a}$ is measurable from it to $(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}))$: the preimage of $A\in\mathcal{B}(\mathbb{R}^{d})$ is $(\mathrm{pr}_{1})^{-1}(A)\cap E_{a}$, a Borel subset of $E_{a}$. Let $\alpha_{a}$ be $w_{a}^{-1}$ times the image measure of $\pi_{12}|_{E_{a}}$ under that restriction, a measure by Image Measures and Measures with a Density: Measures, Integrals and Change of Variables and by Nonnegative Combinations of Two Finite Measures, and the Average of Two Couplings (i) applied with the nonnegative scalars $s=w_{a}^{-1}$ and $t=0$; explicitly

$$\alpha_{a}(A)=w_{a}^{-1}\,\pi_{12}\bigl((\mathrm{pr}_{1})^{-1}(A)\cap E_{a}\bigr)\qquad(A\in\mathcal{B}(\mathbb{R}^{d})).$$

Taking $A=\mathbb{R}^{d}$ gives $\alpha_{a}(\mathbb{R}^{d})=w_{a}^{-1}\pi_{12}(E_{a})=1$, so $\alpha_{a}\in\mathcal{P}(\mathbb{R}^{d})$. Symmetrically, using $G_{a}$, the restriction of $\mathrm{pr}_{2}$ and $\pi_{23}$, define $\beta_{a}\in\mathcal{P}(\mathbb{R}^{d})$ with

$$\beta_{a}(B)=w_{a}^{-1}\,\pi_{23}\bigl(G_{a}\cap(\mathrm{pr}_{2})^{-1}(B)\bigr)\qquad(B\in\mathcal{B}(\mathbb{R}^{d})).$$

The candidate measure. For $a\in F'$ let $\Psi_{a}:\mathbb{R}^{d+d}\to\mathbb{R}^{3d}$ be the map $u\mapsto\iota_{3}(\mathrm{pr}_{1}(u),a,\mathrm{pr}_{2}(u))$. It is Borel: the constant map with value $a$ is Borel, being continuous, so $\Psi_{a}$ is a pairing of pairings of Borel maps, Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (ii). Let $\alpha_{a}\boxtimes\beta_{a}\in\mathcal{P}(\mathbb{R}^{d+d})$ be the product measure of Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (iii), and put

$$\sigma=\sum_{a\in F'}w_{a}\,(\Psi_{a})_{\#}(\alpha_{a}\boxtimes\beta_{a}),$$

a finite nonnegative combination of probability measures on $(\mathbb{R}^{3d},\mathcal{B}(\mathbb{R}^{3d}))$, which is a measure by Nonnegative Combinations of Two Finite Measures, and the Average of Two Couplings (i) applied repeatedly, once for each element of the finite set $F'$ beyond the first. Its total mass is $\sum_{a\in F'}w_{a}=\rho(F')=1$, so $\sigma\in\mathcal{P}(\mathbb{R}^{3d})$.

The first pairwise marginal. Let $C\in\mathcal{B}(\mathbb{R}^{d+d})$ and let $a\in F'$. For $u\in\mathbb{R}^{d+d}$, claim 1 gives $\mathrm{q}_{1}(\Psi_{a}(u))=\mathrm{pr}_{1}(u)$ and $\mathrm{q}_{2}(\Psi_{a}(u))=a$, so

$$(\mathrm{q}_{1},\mathrm{q}_{2})\bigl(\Psi_{a}(u)\bigr)=\iota^{d,d}\bigl(\mathrm{pr}_{1}(u),a\bigr).$$

Put $C_{a}=\{x\in\mathbb{R}^{d}:\iota^{d,d}(x,a)\in C\}$, the preimage of $C$ under the Borel map $x\mapsto\iota^{d,d}(x,a)$ and hence Borel. Then $\Psi_{a}^{-1}((\mathrm{q}_{1},\mathrm{q}_{2})^{-1}(C))=(\mathrm{pr}_{1})^{-1}(C_{a})$, and by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (iii), the image of $\alpha_{a}\boxtimes\beta_{a}$ under $\mathrm{pr}_{1}$ being $\alpha_{a}$,

$$(\alpha_{a}\boxtimes\beta_{a})\Bigl(\Psi_{a}^{-1}\bigl((\mathrm{q}_{1},\mathrm{q}_{2})^{-1}(C)\bigr)\Bigr)=\alpha_{a}(C_{a}).$$

Now $w_{a}\,\alpha_{a}(C_{a})=\pi_{12}((\mathrm{pr}_{1})^{-1}(C_{a})\cap E_{a})$, and this set is $C\cap E_{a}$: a point $u$ lies in it exactly when $\mathrm{pr}_{2}(u)=a$ and $\iota^{d,d}(\mathrm{pr}_{1}(u),a)\in C$, and for such a $u$ one has $u=\iota^{d,d}(\mathrm{pr}_{1}(u),\mathrm{pr}_{2}(u))=\iota^{d,d}(\mathrm{pr}_{1}(u),a)$ by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (i), so that the second condition reads $u\in C$. Summing over $a\in F'$ and using the definition of $\sigma$ and of a push-forward,

$$(\mathrm{q}_{1},\mathrm{q}_{2})_{\#}\sigma(C)=\sum_{a\in F'}\pi_{12}(C\cap E_{a})=\pi_{12}\Bigl(C\cap(\mathrm{pr}_{2})^{-1}(F')\Bigr)=\pi_{12}(C),$$

the middle equality by claim 1 of Basic Properties of a Measure, the sets $C\cap E_{a}$ being pairwise disjoint with union $C\cap(\mathrm{pr}_{2})^{-1}(F')$, and the last as follows. By Couplings of Two Probability Measures on Euclidean Space and Their Quadratic Cost (i), $\pi_{12}((\mathrm{pr}_{2})^{-1}(\mathbb{R}^{d}\setminus F'))=\rho(\mathbb{R}^{d}\setminus F')$, which is $0$ by claim 3 of Basic Properties of a Measure applied to the Borel set $F'$ with $\rho(F')=1$ and $\rho(\mathbb{R}^{d})=1$. Since $C\setminus(\mathrm{pr}_{2})^{-1}(F')$ is contained in $(\mathrm{pr}_{2})^{-1}(\mathbb{R}^{d}\setminus F')$, it too is $\pi_{12}$-null by claim 2 of Basic Properties of a Measure, so claim 3 of that lemma gives $\pi_{12}(C\cap(\mathrm{pr}_{2})^{-1}(F'))=\pi_{12}(C)$.

The second pairwise marginal. Symmetrically, $\mathrm{q}_{2}(\Psi_{a}(u))=a$ and $\mathrm{q}_{3}(\Psi_{a}(u))=\mathrm{pr}_{2}(u)$, so $(\mathrm{q}_{2},\mathrm{q}_{3})(\Psi_{a}(u))=\iota^{d,d}(a,\mathrm{pr}_{2}(u))$. With $C^{a}=\{y\in\mathbb{R}^{d}:\iota^{d,d}(a,y)\in C\}$ one gets $\Psi_{a}^{-1}((\mathrm{q}_{2},\mathrm{q}_{3})^{-1}(C))=(\mathrm{pr}_{2})^{-1}(C^{a})$, whose $\alpha_{a}\boxtimes\beta_{a}$-measure is $\beta_{a}(C^{a})$ by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (iii); then $w_{a}\beta_{a}(C^{a})=\pi_{23}(G_{a}\cap(\mathrm{pr}_{2})^{-1}(C^{a}))=\pi_{23}(C\cap G_{a})$ by the same computation with the roles of the two coordinates exchanged, and summing over $a\in F'$ gives $(\mathrm{q}_{2},\mathrm{q}_{3})_{\#}\sigma=\pi_{23}$, the passage from $\pi_{23}(C\cap(\mathrm{pr}_{1})^{-1}(F'))$ to $\pi_{23}(C)$ being justified exactly as in the previous paragraph, now from $\pi_{23}((\mathrm{pr}_{1})^{-1}(\mathbb{R}^{d}\setminus F'))=\rho(\mathbb{R}^{d}\setminus F')=0$.