Proof of Gluing Two Couplings over a Common Middle Marginal, and the Composite Coupling

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.

Step 0: two facts about the concatenation maps. For $x,x',y,y'\in\mathbb{R}^{d}$, claims 2 and 3 of Concatenation Identifies a Product of Euclidean Spaces with a Euclidean Space give $\iota^{d,d}(x,y)-\iota^{d,d}(x',y')=\iota^{d,d}(x-x',y-y')$ and

$$\bigl\lVert\iota^{d,d}(x-x',y-y')\bigr\rVert^{2}=\lVert x-x'\rVert^{2}+\lVert y-y'\rVert^{2}.$$

Applying this twice to the definition of $\iota_{3}$ in Pairings of Borel Maps as Couplings, and Gluing Two Couplings over a Finitely Supported Middle Marginal (i) gives, for $w,w'\in\mathbb{R}^{3d}$,

$$\lVert w-w'\rVert^{2}=\sum_{i=1}^{3}\bigl\lVert\mathrm{q}_{i}(w)-\mathrm{q}_{i}(w')\bigr\rVert^{2},$$

using the representation $w=\iota_{3}(\mathrm{q}_{1}(w),\mathrm{q}_{2}(w),\mathrm{q}_{3}(w))$ of that clause. In particular, taking $w'$ to be the origin, $\lVert w\rVert^{2}=\sum_{i=1}^{3}\lVert\mathrm{q}_{i}(w)\rVert^{2}$; and, discarding the term $i=3$, which is nonnegative,

$$\bigl\lVert(\mathrm{q}_{1},\mathrm{q}_{2})(w)-(\mathrm{q}_{1},\mathrm{q}_{2})(w')\bigr\rVert^{2}=\bigl\lVert\mathrm{q}_{1}(w)-\mathrm{q}_{1}(w')\bigr\rVert^{2}+\bigl\lVert\mathrm{q}_{2}(w)-\mathrm{q}_{2}(w')\bigr\rVert^{2}\le\lVert w-w'\rVert^{2},$$

so that $(\mathrm{q}_{1},\mathrm{q}_{2})$ is Lipschitz with constant $1$ for the Euclidean distances, by claim 2 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square and claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field. The same holds for $(\mathrm{q}_{2},\mathrm{q}_{3})$ and for $(\mathrm{q}_{1},\mathrm{q}_{3})$.

Consequently, if $f:\mathbb{R}^{d+d}\to\mathbb{R}$ is Lipschitz with a constant $L$, then $f\circ(\mathrm{q}_{1},\mathrm{q}_{2})$ is continuous on $\mathbb{R}^{3d}$: given a positive $\varepsilon\in\mathbb{R}$, the number $\delta=\varepsilon\,(L+1)^{-1}$ is positive by claims 5, 6 and 7 of Elementary Order Arithmetic in an Ordered Field, and for $w,w'$ with $d_{E}(w,w')<\delta$ one has

$$\bigl|f\bigl((\mathrm{q}_{1},\mathrm{q}_{2})(w)\bigr)-f\bigl((\mathrm{q}_{1},\mathrm{q}_{2})(w')\bigr)\bigr|\le L\,d_{E}\bigl((\mathrm{q}_{1},\mathrm{q}_{2})(w),(\mathrm{q}_{1},\mathrm{q}_{2})(w')\bigr)\le L\,d_{E}(w,w')<\varepsilon,$$

by the Lipschitz property of $f$, the Lipschitz bound just proved, claim 5 of Elementary Arithmetic in an Ordered Field and $L\,\delta<\varepsilon$.

Step 1: quantisation of the middle marginal. Let $k\in\mathbb{N}$ and let $\iota(k)$ be its image in $\mathbb{R}$ under the canonical map of The Canonical Map from the Natural Numbers to a Field, a positive real number by claim 3 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field, so that $\iota(k)^{-1}$ is positive by claim 7 of Elementary Order Arithmetic in an Ordered Field. By Couplings on Euclidean Space: Product Coupling, Swap, Finiteness of the Cost, Push-Forward Couplings, Modifying One Marginal, Quantisation, Gluing over a Finitely Supported Measure, and the Lipschitz Bound (vi) applied to $\rho\in\mathcal{P}_{2}(\mathbb{R}^{d})$ and to $\iota(k)^{-1}$ there is a Borel map $R_{k}:\mathbb{R}^{d}\to\mathbb{R}^{d}$ whose image is a finite set and which satisfies

$$\int_{\mathbb{R}^{d}}\lVert R_{k}(y)-y\rVert^{2}\,\rho(dy)\le\iota(k)^{-2} .$$

Put $\rho_{k}=(R_{k})_{\#}\rho$. By the same clause $\rho_{k}(\mathbb{R}^{d}\setminus R_{k}(\mathbb{R}^{d}))=0$ with $R_{k}(\mathbb{R}^{d})$ finite. By the inequality $\lVert x\rVert^{2}\le2\lVert y\rVert^{2}+2\lVert x-y\rVert^{2}$ of Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (iv), applied pointwise with $x=R_{k}(y)$, and by claim 1 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion,

$$\int_{\mathbb{R}^{d}}\lVert R_{k}\rVert^{2}\,d\rho\le2\,M_{2}(\rho)+2\,\iota(k)^{-2}\le2\,M_{2}(\rho)+2 ,$$

the last step because $1\le\iota(k)$ by claim 6 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field together with claim 4 of Properties of the Order on the Natural Numbers, whence $\iota(k)^{-1}\le1$ and $\iota(k)^{-2}\le1$ by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field. So $\rho_{k}\in\mathcal{P}_{2}(\mathbb{R}^{d})$ with $M_{2}(\rho_{k})\le2M_{2}(\rho)+2$, by The Optimal Map as a Square-Integrable Vector Field: Integrability, Transport Cost and Uniqueness of the Class (i).

By Couplings on Euclidean Space: Product Coupling, Swap, Finiteness of the Cost, Push-Forward Couplings, Modifying One Marginal, Quantisation, Gluing over a Finitely Supported Measure, and the Lipschitz Bound (v), applied once with $T=R_{k}$ to $\pi_{12}$ and once with $S=R_{k}$ to $\pi_{23}$,

$$\pi^{k}_{12}=(\mathrm{pr}_{1},R_{k}\circ\mathrm{pr}_{2})_{\#}\pi_{12}\in\Pi(\mu,\rho_{k}),\qquad\pi^{k}_{23}=(R_{k}\circ\mathrm{pr}_{1},\mathrm{pr}_{2})_{\#}\pi_{23}\in\Pi(\rho_{k},\nu).$$

Since $\rho_{k}$ vanishes off a finite set, Pairings of Borel Maps as Couplings, and Gluing Two Couplings over a Finitely Supported Middle Marginal (iii) supplies $\sigma_{k}\in\mathcal{P}(\mathbb{R}^{3d})$ with

$$(\mathrm{q}_{1},\mathrm{q}_{2})_{\#}\sigma_{k}=\pi^{k}_{12},\qquad(\mathrm{q}_{2},\mathrm{q}_{3})_{\#}\sigma_{k}=\pi^{k}_{23}.$$

Step 2: a uniform second-moment bound, and a weak limit. Fix $k$. By Pairings of Borel Maps as Couplings, and Gluing Two Couplings over a Finitely Supported Middle Marginal (ii), read with $3d$ in place of $m$ and $d$ in place of $n$, the measure $(\mathrm{q}_{1},\mathrm{q}_{2})_{\#}\sigma_{k}$ is a coupling of $(\mathrm{q}_{1})_{\#}\sigma_{k}$ and $(\mathrm{q}_{2})_{\#}\sigma_{k}$; it is also $\pi^{k}_{12}$, a coupling of $\mu$ and $\rho_{k}$. Since by Couplings of Two Probability Measures on Euclidean Space and Their Quadratic Cost (i) the two marginals of a coupling are its push-forwards under the two coordinate projections, hence determined by it,

$$(\mathrm{q}_{1})_{\#}\sigma_{k}=\mu,\qquad(\mathrm{q}_{2})_{\#}\sigma_{k}=\rho_{k},$$

and the same argument applied to $(\mathrm{q}_{2},\mathrm{q}_{3})_{\#}\sigma_{k}=\pi^{k}_{23}\in\Pi(\rho_{k},\nu)$ gives $(\mathrm{q}_{3})_{\#}\sigma_{k}=\nu$. By step 0, $\lVert w\rVert^{2}=\sum_{i=1}^{3}\lVert\mathrm{q}_{i}(w)\rVert^{2}$, so claim 1 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion, the change-of-variables formula of Probability Measures on Euclidean Space and Random Vectors: Standing Notation (iv) and The Second Moment of a Probability Measure on Euclidean Space and the Probability Measures with Finite Second Moment (i) give

$$M_{2}(\sigma_{k})=M_{2}(\mu)+M_{2}(\rho_{k})+M_{2}(\nu)\le M_{2}(\mu)+2M_{2}(\rho)+2+M_{2}(\nu)=:R^{*},$$

a real number not depending on $k$. By Tightness from Bounded Second Moments, and Tightness of the Couplings of Two Measures with Finite Second Moment (i), read with $3d$ in place of the dimension $m$ there and with $\mathcal{M}$ the set of the terms of the sequence $(\sigma_{k})_{k\in\mathbb{N}}$, that sequence is tight in $(\mathbb{R}^{3d},d_{E})$; so by Prokhorov's Theorem on Euclidean Space: a Tight Sequence of Probability Measures Has a Weakly Convergent Subsequence, again with $3d$ in place of $m$, there are a strictly increasing $(k_{j})_{j\in\mathbb{N}}$ in $\mathbb{N}$ and $\sigma\in\mathcal{P}(\mathbb{R}^{3d})$ such that $(\sigma_{k_{j}})_{j\in\mathbb{N}}$ converges weakly to $\sigma$.

Step 3: identifying the pairwise marginals of $\sigma$. Let $f:\mathbb{R}^{d+d}\to\mathbb{R}$ be bounded and Lipschitz, with constant $L$. The composite $f\circ(\mathrm{q}_{1},\mathrm{q}_{2})$ is bounded and continuous on $\mathbb{R}^{3d}$ by step 0 and Semicontinuity and Continuity Under Composition with a Continuous Map, so by weak convergence and the change-of-variables formula of Probability Measures on Euclidean Space and Random Vectors: Standing Notation (iv),

$$\int_{\mathbb{R}^{d+d}}f\,d\pi^{k_{j}}_{12}=\int_{\mathbb{R}^{3d}}f\circ(\mathrm{q}_{1},\mathrm{q}_{2})\,d\sigma_{k_{j}}\ \longrightarrow\ \int_{\mathbb{R}^{3d}}f\circ(\mathrm{q}_{1},\mathrm{q}_{2})\,d\sigma=\int_{\mathbb{R}^{d+d}}f\,d\bigl((\mathrm{q}_{1},\mathrm{q}_{2})_{\#}\sigma\bigr)$$

as $j\to\infty$, the integrals being defined because a bounded Borel function is integrable against a probability measure by Probability Measures on Euclidean Space and Random Vectors: Standing Notation (iii).

On the other hand the same integrals converge to $\int f\,d\pi_{12}$. Indeed, let $\Gamma_{k}=(\mathrm{id}_{d+d},(\mathrm{pr}_{1},R_{k}\circ\mathrm{pr}_{2}))_{\#}\pi_{12}$, where $\mathrm{id}_{d+d}$ is the identity map of $\mathbb{R}^{d+d}$. By Pairings of Borel Maps as Couplings, and Gluing Two Couplings over a Finitely Supported Middle Marginal (ii), read with $d+d$ in place of both $m$ and $n$, $\Gamma_{k}$ is a coupling of $\pi_{12}$ and $\pi^{k}_{12}$, with

$$I(\Gamma_{k})=\int_{\mathbb{R}^{d+d}}\bigl\lVert u-(\mathrm{pr}_{1}(u),R_{k}(\mathrm{pr}_{2}(u)))\bigr\rVert^{2}\,\pi_{12}(du)=\int_{\mathbb{R}^{d+d}}\bigl\lVert\mathrm{pr}_{2}(u)-R_{k}(\mathrm{pr}_{2}(u))\bigr\rVert^{2}\,\pi_{12}(du),$$

the second equality by the identity of step 0 for $\iota^{d,d}$, the first coordinates of the two points agreeing. By the change-of-variables formula and $(\mathrm{pr}_{2})_{\#}\pi_{12}=\rho$, this equals $\int_{\mathbb{R}^{d}}\lVert y-R_{k}(y)\rVert^{2}\rho(dy)$, which is at most $\iota(k)^{-2}$ by step 1 (the two integrands agreeing pointwise by claim 5 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square with the scalar $-1$). Hence, by Couplings on Euclidean Space: Product Coupling, Swap, Finiteness of the Cost, Push-Forward Couplings, Modifying One Marginal, Quantisation, Gluing over a Finitely Supported Measure, and the Lipschitz Bound (viii) read in the dimension $d+d$,

$$\Bigl|\int_{\mathbb{R}^{d+d}}f\,d\pi_{12}-\int_{\mathbb{R}^{d+d}}f\,d\pi^{k}_{12}\Bigr|\le L\,\sqrt{I(\Gamma_{k})}\le L\,\iota(k)^{-1},$$

using claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field for the last step. The right-hand side converges to $0$ as $k\to\infty$ by The Archimedean Property of the Real Numbers, so the sequence $(\int f\,d\pi^{k}_{12})_{k\in\mathbb{N}}$ converges to $\int f\,d\pi_{12}$ by Limit of a Sequence of Real Numbers, and so does the subsequence indexed by $(k_{j})$ by A Subsequence of a Convergent Sequence Has the Same Limit.

By the uniqueness of limits of real sequences, Uniqueness of Limits in a Metric Space,

$$\int_{\mathbb{R}^{d+d}}f\,d\bigl((\mathrm{q}_{1},\mathrm{q}_{2})_{\#}\sigma\bigr)=\int_{\mathbb{R}^{d+d}}f\,d\pi_{12}$$

for every bounded Lipschitz $f:\mathbb{R}^{d+d}\to\mathbb{R}$; both measures being finite Borel measures on the metric space $(\mathbb{R}^{d+d},d_{E})$, claim 1 of Lipschitz Test Functions Determine a Finite Borel Measure, and Uniqueness of Weak Limits gives $(\mathrm{q}_{1},\mathrm{q}_{2})_{\#}\sigma=\pi_{12}$. The identical argument with $(\mathrm{q}_{2},\mathrm{q}_{3})$, with $\Gamma'_{k}=(\mathrm{id}_{d+d},(R_{k}\circ\mathrm{pr}_{1},\mathrm{pr}_{2}))_{\#}\pi_{23}$ and with $(\mathrm{pr}_{1})_{\#}\pi_{23}=\rho$, gives $(\mathrm{q}_{2},\mathrm{q}_{3})_{\#}\sigma=\pi_{23}$. This proves claim 1.

Claim 2. Let $\sigma$ be a gluing of $\pi_{12}$ and $\pi_{23}$. The argument of step 2, applied to $\sigma$ with $\pi_{12}\in\Pi(\mu,\rho)$ and $\pi_{23}\in\Pi(\rho,\nu)$ in place of $\pi^{k}_{12}$ and $\pi^{k}_{23}$, gives $(\mathrm{q}_{1})_{\#}\sigma=\mu$, $(\mathrm{q}_{2})_{\#}\sigma=\rho$ and $(\mathrm{q}_{3})_{\#}\sigma=\nu$. As in step 2, the change-of-variables formula of Probability Measures on Euclidean Space and Random Vectors: Standing Notation (iv) applied to $\mathrm{q}_{1}$ and to the nonnegative Borel function $x\mapsto\lVert x\rVert^{2}$, together with The Second Moment of a Probability Measure on Euclidean Space and the Probability Measures with Finite Second Moment (i), gives $\int_{\mathbb{R}^{3d}}\lVert\mathrm{q}_{1}\rVert^{2}\,d\sigma=M_{2}(\mu)<\infty$, and likewise $\int\lVert\mathrm{q}_{2}\rVert^{2}\,d\sigma=M_{2}(\rho)$ and $\int\lVert\mathrm{q}_{3}\rVert^{2}\,d\sigma=M_{2}(\nu)$, all finite; so the three classes belong to $L^{2}(\sigma;\mathbb{R}^{d})$ with the stated squared norms, by Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation (ii).

For the two cost identities, $(\mathrm{q}_{1},\mathrm{q}_{2})_{\#}\sigma=\pi_{12}$ and the change-of-variables formula of Probability Measures on Euclidean Space and Random Vectors: Standing Notation (iv) applied to the nonnegative Borel integrand of Couplings of Two Probability Measures on Euclidean Space and Their Quadratic Cost (ii) give

$$I(\pi_{12})=\int_{\mathbb{R}^{3d}}\bigl\lVert\mathrm{q}_{1}(w)-\mathrm{q}_{2}(w)\bigr\rVert^{2}\,\sigma(dw)=\lVert\mathrm{q}_{1}-\mathrm{q}_{2}\rVert_{\sigma}^{2},$$

using Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (i) to evaluate the integrand at $(\mathrm{q}_{1},\mathrm{q}_{2})(w)$ and Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation (ii) for the last equality. The same computation with $(\mathrm{q}_{2},\mathrm{q}_{3})$ gives $\lVert\mathrm{q}_{2}-\mathrm{q}_{3}\rVert_{\sigma}^{2}=I(\pi_{23})$.

Claim 3. By Pairings of Borel Maps as Couplings, and Gluing Two Couplings over a Finitely Supported Middle Marginal (ii), read with $3d$ in place of $m$ and $d$ in place of $n$ and with $\mathrm{q}_{1}$, $\mathrm{q}_{3}$ as the two Borel maps, $(\mathrm{q}_{1},\mathrm{q}_{3})_{\#}\sigma$ belongs to $\Pi((\mathrm{q}_{1})_{\#}\sigma,(\mathrm{q}_{3})_{\#}\sigma)=\Pi(\mu,\nu)$ and

$$I\bigl((\mathrm{q}_{1},\mathrm{q}_{3})_{\#}\sigma\bigr)=\int_{\mathbb{R}^{3d}}\lVert\mathrm{q}_{1}-\mathrm{q}_{3}\rVert^{2}\,d\sigma=\lVert\mathrm{q}_{1}-\mathrm{q}_{3}\rVert_{\sigma}^{2},$$

the last equality by Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation (ii), the class of $\mathrm{q}_{1}-\mathrm{q}_{3}$ lying in $L^{2}(\sigma;\mathbb{R}^{d})$ by claim 2 and the vector space structure of that space.

Finally, in the real inner product space $L^{2}(\sigma;\mathbb{R}^{d})$ the triangle inequality, claim 1 of The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity, applied to $\mathrm{q}_{1}-\mathrm{q}_{3}=(\mathrm{q}_{1}-\mathrm{q}_{2})+(\mathrm{q}_{2}-\mathrm{q}_{3})$, gives

$$\lVert\mathrm{q}_{1}-\mathrm{q}_{3}\rVert_{\sigma}\le\lVert\mathrm{q}_{1}-\mathrm{q}_{2}\rVert_{\sigma}+\lVert\mathrm{q}_{2}-\mathrm{q}_{3}\rVert_{\sigma}=\sqrt{I(\pi_{12})}+\sqrt{I(\pi_{23})},$$

the equality by claim 2 and Existence and Uniqueness of the Nonnegative Square Root, the norms being nonnegative with the stated squares. Since $\lVert\mathrm{q}_{1}-\mathrm{q}_{3}\rVert_{\sigma}$ is the nonnegative square root of $I((\mathrm{q}_{1},\mathrm{q}_{3})_{\#}\sigma)$, again by Existence and Uniqueness of the Nonnegative Square Root, this is the asserted inequality.