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

In the setting of Probability Measures on Euclidean Space and Random Vectors: Standing Notation, let $d\in\mathbb{N}$ satisfy $1\le d$ and let $\mu,\rho,\nu\in\mathcal{P}(\mathbb{R}^{d})$, with $\Pi(\mu,\rho)$ and $\Pi(\rho,\nu)$ the sets of their couplings. As in Probability Measures on Euclidean Space and Random Vectors: Standing Notation (v), $\iota^{q,p}$ is the concatenation map and $\mathrm{pr}^{q,p}_{1}$, $\mathrm{pr}^{q,p}_{2}$ are the coordinate projections, with $\mathrm{pr}_{1}=\mathrm{pr}^{d,d}_{1}$ and $\mathrm{pr}_{2}=\mathrm{pr}^{d,d}_{2}$; push-forwards are those of Probability Measures on Euclidean Space and Random Vectors: Standing Notation (iv), and finite subsets of $\mathbb{R}^{d}$ are Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (v). In this statement the letters $q$ and $p$ are used only as the dimension superscripts just named.

1. (The coordinate maps of the threefold product)(i) Write $\mathbb{R}^{3d}$ for the Euclidean space $\mathbb{R}^{(d+d)+d}$, and let

$$\mathrm{q}_{1}=\mathrm{pr}_{1}\circ\mathrm{pr}^{d+d,d}_{1},\qquad \mathrm{q}_{2}=\mathrm{pr}_{2}\circ\mathrm{pr}^{d+d,d}_{1},\qquad \mathrm{q}_{3}=\mathrm{pr}^{d+d,d}_{2}$$

be maps from $\mathbb{R}^{3d}$ to $\mathbb{R}^{d}$, called the coordinate maps of $\mathbb{R}^{3d}$. Each is Borel, a composition of Borel maps being Borel by Probability Measures on Euclidean Space and Random Vectors: Standing Notation (ii), and therefore the pairings $(\mathrm{q}_{1},\mathrm{q}_{2})$, $(\mathrm{q}_{2},\mathrm{q}_{3})$ and $(\mathrm{q}_{1},\mathrm{q}_{3})$ are Borel maps from $\mathbb{R}^{3d}$ to $\mathbb{R}^{d+d}$ by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (ii). For $x,y,z\in\mathbb{R}^{d}$ we write

$$\iota_{3}(x,y,z)=\iota^{d+d,d}\bigl(\iota^{d,d}(x,y),z\bigr)\in\mathbb{R}^{3d};$$

by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (i) one has $\mathrm{q}_{1}(\iota_{3}(x,y,z))=x$, $\mathrm{q}_{2}(\iota_{3}(x,y,z))=y$ and $\mathrm{q}_{3}(\iota_{3}(x,y,z))=z$, and every $w\in\mathbb{R}^{3d}$ equals $\iota_{3}(\mathrm{q}_{1}(w),\mathrm{q}_{2}(w),\mathrm{q}_{3}(w))$, so that $\iota_{3}$ is a bijection of $\mathbb{R}^{d}\times\mathbb{R}^{d}\times\mathbb{R}^{d}$ onto $\mathbb{R}^{3d}$.

2. (Pairings of Borel maps as couplings)(ii) Let $m,n\in\mathbb{N}$ satisfy $1\le m$ and $1\le n$, let $\lambda\in\mathcal{P}(\mathbb{R}^{m})$ and let $S,T:\mathbb{R}^{m}\to\mathbb{R}^{n}$ be Borel, so that the pairing $(S,T):\mathbb{R}^{m}\to\mathbb{R}^{n+n}$ is Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (ii) and the function $s\mapsto\lVert S(s)-T(s)\rVert^{2}$ is Borel and nonnegative by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (iv). Then

$$(S,T)_{\#}\lambda\in\Pi\bigl(S_{\#}\lambda,\,T_{\#}\lambda\bigr),\qquad I\bigl((S,T)_{\#}\lambda\bigr)=\int_{\mathbb{R}^{m}}\lVert S-T\rVert^{2}\,d\lambda$$

in $[0,\infty]$, where $\Pi$ and the quadratic cost $I$ are read in the dimension $n$ and $\lVert S-T\rVert^{2}$ denotes the function just named.

3. (Gluing over a finitely supported middle marginal)(iii) Suppose that $\rho(\mathbb{R}^{d}\setminus F)=0$ for some finite set $F\subseteq\mathbb{R}^{d}$, and let $\pi_{12}\in\Pi(\mu,\rho)$ and $\pi_{23}\in\Pi(\rho,\nu)$. Then there is $\sigma\in\mathcal{P}(\mathbb{R}^{3d})$ with

$$(\mathrm{q}_{1},\mathrm{q}_{2})_{\#}\sigma=\pi_{12},\qquad(\mathrm{q}_{2},\mathrm{q}_{3})_{\#}\sigma=\pi_{23}.$$