Proof of Gluing Two Couplings on a Hilbert Space over a Common Middle Marginal, and the Triangle Inequalities for the Quadratic and Noise Costs

Each result cited is universally quantified over the data in its own statement.

Conventions. "Change of variables" is claim 2 of Image Measures, Measures with Densities, and Change of Variables, for the push-forwards of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §pushforward and of the statement; "the integral theorem" is Linearity and Monotonicity of the Lebesgue Integral. For measurable maps $G$ and $H$ the image measures of claim 1 of Image Measures, Measures with Densities, and Change of Variables satisfy $(H\circ G)_{\#}\tau=H_{\#}(G_{\#}\tau)$, both sides assigning to $B$ the value $\tau(G^{-1}(H^{-1}(B)))$, and $G_{\#}\tau$ has the total mass of $\tau$. For maps $S,T$ into $X$ one has $\pi_{1}\circ(S,T)=S$ and $\pi_{2}\circ(S,T)=T$, and a pair is determined by its coordinates, by The Product of Two Real Inner Product Spaces §notation; in particular $\pi_{1}\circ(q_{1},q_{2})=q_{1}=\pi_{1}\circ(q_{1},q_{3})$, $\pi_{2}\circ(q_{1},q_{2})=q_{2}=\pi_{1}\circ(q_{2},q_{3})$ and $\pi_{2}\circ(q_{2},q_{3})=q_{3}=\pi_{2}\circ(q_{1},q_{3})$, and $w=((q_{1}w,q_{2}w),q_{3}w)$, so $(q_{1},q_{2})(w)$ is the first coordinate of $w$ in $(X\times X)\times X$. Measure-theoretic facts are those of Basic Properties of a Measure, cited by anchor. Applying Properties of the Product of Two Real Inner Product Spaces §metric to $X\times X$ and then to $X_{(3)}$, for $w=((u,v),s)$ and $w'=((u',v'),s')$,

$$d(w,w')^{2}=|u-u'|^{2}+|v-v'|^{2}+|s-s'|^{2}. \tag{M}$$

Hence $(q_{1},q_{2})$ and $(q_{2},q_{3})$ are Lipschitz with constant $1$ from $(X_{(3)},d)$ to $(X\times X,d)$, so continuous by A Lipschitz Map is Uniformly Continuous. For $p\in X$ let $\psi_{p}:X\times X\to X_{(3)}$, $\psi_{p}(x,s)=((x,p),s)$, and $\iota_{p},\iota'_{p}:X\to X\times X$, $\iota_{p}(x)=(x,p)$, $\iota'_{p}(s)=(p,s)$. By (M) and Properties of the Product of Two Real Inner Product Spaces §metric these maps preserve distances, so they are continuous by A Lipschitz Map is Uniformly Continuous and Borel by claim 3 of Borel Measurability and Bounded Integration on a Metric Space.

Two real inequalities. (E1) For real $r,s\ge0$ and $t>0$, $(r+s)^{2}\le(1+t)r^{2}+(1+t^{-1})s^{2}$, since the difference is $(tr-s)^{2}/t\ge0$. (E2) Let $A,B,C\ge0$ be real with $C\le(1+t)A+(1+t^{-1})B$ for every real $t>0$. Then $\sqrt{C}\le\sqrt{A}+\sqrt{B}$ (square roots of Existence and Uniqueness of the Nonnegative Square Root). Indeed, if $A>0$ and $B>0$, the choice $t=\sqrt{B}/\sqrt{A}$ gives $C\le A+2\sqrt{A}\sqrt{B}+B=(\sqrt{A}+\sqrt{B})^{2}$. If $A=0$, then $C\le B$: otherwise $C>B$, and $t=1$ excludes $B=0$, while for $B>0$ the choice $t=2B/(C-B)$ gives $C\le B+(C-B)/2<C$. If $B=0$, then likewise $C\le A$: otherwise $C>A$, $A=0$ is excluded by the case just treated, and for $A>0$ the choice $t=(C-A)/(2A)$ gives $C\le A+(C-A)/2<C$. In all cases $C\le(\sqrt{A}+\sqrt{B})^{2}$, and claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field gives the assertion.

Step 1 (gluing over a finitely supported middle marginal). Let $\lambda'\in\mathcal{P}(X)$ with $\lambda'(X\setminus F)=0$ for a finite $F\subseteq X$, and let $\pi'_{12}\in\Pi(\mu,\lambda')$ and $\pi'_{23}\in\Pi(\lambda',\nu)$. We construct $\sigma'\in\mathcal{P}(X_{(3)})$ with $(q_{1},q_{2})_{\#}\sigma'=\pi'_{12}$ and $(q_{2},q_{3})_{\#}\sigma'=\pi'_{23}$. Finite subsets of $X$ and their complements are Borel, as recorded in the statement of Couplings on a Hilbert Space: Product Coupling, Swap, Finiteness of the Cost, Push-Forward Couplings, Modifying One Marginal, Quantisation, Gluing over a Finitely Supported Measure, the Lipschitz Bound and the Moment Bound. Let $F_{+}=\{p\in F:0<\lambda'(\{p\})\}$, finite by claim 3 of Basic Properties of Finite Sets, and $r_{p}=\lambda'(\{p\})$ for $p\in F_{+}$. The finite set $F\setminus F_{+}$ is either empty, in which case $\lambda'(F\setminus F_{+})=0$ trivially, or a finite disjoint union of $\lambda'$-null singletons, so that $\lambda'(F\setminus F_{+})=0$ by Basic Properties of a Measure §additivity; whence $\lambda'(X\setminus F_{+})=\lambda'(X\setminus F)+\lambda'(F\setminus F_{+})=0$ by the same claim, $\lambda'(F_{+})=1$ by Basic Properties of a Measure §differences, and $F_{+}\neq\varnothing$, since $\lambda'(F_{+})=1\neq0=\lambda'(\varnothing)$. By the marginal conditions of Couplings of Two Borel Probability Measures on a Hilbert Space and Their Quadratic Cost §coupling,

$$\pi'_{12}\bigl(\pi_{2}^{-1}(X\setminus F_{+})\bigr)=0=\pi'_{23}\bigl(\pi_{1}^{-1}(X\setminus F_{+})\bigr). \tag{N}$$

For $p\in F_{+}$ and $A,C\in\mathcal{B}(X)$ put $\alpha_{p}(A)=\pi'_{12}(\pi_{1}^{-1}(A)\cap\pi_{2}^{-1}(\{p\}))$ and $\beta_{p}(C)=\pi'_{23}(\pi_{1}^{-1}(\{p\})\cap\pi_{2}^{-1}(C))$. Preimages and intersection with a fixed Borel set preserve countable disjoint unions, so $\alpha_{p},\beta_{p}$ are Borel measures on $(X,d)$ (Measure, Measure Space, and Probability Measure), with $\alpha_{p}(X)=\lambda'(\{p\})=r_{p}=\beta_{p}(X)$ by the marginal conditions. By Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §product-measure the product measure $\theta_{p}=\alpha_{p}\otimes\beta_{p}$ is a Borel measure on $X\times X$, and $\theta_{p}(A\times C)=\alpha_{p}(A)\beta_{p}(C)$ for $A,C\in\mathcal{B}(X)$ by Existence and Uniqueness of the Product Measure, as $\mathcal{B}(X\times X)=\mathcal{B}(X)\otimes\mathcal{B}(X)$ by Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §product-sigma. Define, for $W\in\mathcal{B}(X_{(3)})$,

$$\sigma'(W)=\sum_{p\in F_{+}}r_{p}^{-1}\,\theta_{p}\bigl(\psi_{p}^{-1}(W)\bigr).$$

Each summand is a positive multiple of the image measure $(\psi_{p})_{\#}\theta_{p}$, and a finite sum of measures with positive real coefficients is a measure (series of nonnegative terms add termwise in $[0,\infty]$), so $\sigma'$ is a Borel measure on $X_{(3)}$, with $\sigma'(X_{(3)})=\sum_{p}r_{p}^{-1}\alpha_{p}(X)\beta_{p}(X)=\sum_{p}\lambda'(\{p\})=\lambda'(F_{+})=1$ by Basic Properties of a Measure §additivity. Thus $\sigma'\in\mathcal{P}(X_{(3)})$.

Let $E\in\mathcal{B}(X\times X)$ and $p\in F_{+}$. Since $(q_{1},q_{2})(\psi_{p}(x,s))=(x,p)$, one has $\psi_{p}^{-1}((q_{1},q_{2})^{-1}(E))=E^{p}\times X$ with $E^{p}=\iota_{p}^{-1}(E)\in\mathcal{B}(X)$, so $\theta_{p}(\psi_{p}^{-1}((q_{1},q_{2})^{-1}(E)))=r_{p}\,\alpha_{p}(E^{p})$; and a pair $z$ with $\pi_{2}(z)=p$ equals $(\pi_{1}(z),p)$, so $\pi_{1}^{-1}(E^{p})\cap\pi_{2}^{-1}(\{p\})=E\cap\pi_{2}^{-1}(\{p\})$. These sets are pairwise disjoint for $p\in F_{+}$ with union $E\cap\pi_{2}^{-1}(F_{+})$, so by Basic Properties of a Measure §additivity, Basic Properties of a Measure §monotone and (N),

$$\bigl((q_{1},q_{2})_{\#}\sigma'\bigr)(E)=\sum_{p\in F_{+}}\alpha_{p}(E^{p})=\pi'_{12}\bigl(E\cap\pi_{2}^{-1}(F_{+})\bigr)=\pi'_{12}(E).$$

Likewise $(q_{2},q_{3})(\psi_{p}(x,s))=(p,s)$, so $\psi_{p}^{-1}((q_{2},q_{3})^{-1}(E))=X\times E_{p}$ with $E_{p}=(\iota'_{p})^{-1}(E)$, its $\theta_{p}$-measure is $r_{p}\beta_{p}(E_{p})$, $\pi_{1}^{-1}(\{p\})\cap\pi_{2}^{-1}(E_{p})=E\cap\pi_{1}^{-1}(\{p\})$, and the same computation with (N) gives $((q_{2},q_{3})_{\#}\sigma')(E)=\pi'_{23}(E)$.

Step 2 (quantisation). Fix $m\in\mathbb{N}$. By Couplings on a Hilbert Space: Product Coupling, Swap, Finiteness of the Cost, Push-Forward Couplings, Modifying One Marginal, Quantisation, Gluing over a Finitely Supported Measure, the Lipschitz Bound and the Moment Bound §quantisation, applied to $\lambda\in\mathcal{P}_{2}(X)$ with $\varepsilon=1/m$, choose a Borel $T_{m}:X\to X$ with finite image $F_{m}=T_{m}(X)$ and $\int_{X}|T_{m}(y)-y|^{2}\,\lambda(dy)\le m^{-2}$; by the same clause $\lambda_{m}=(T_{m})_{\#}\lambda\in\mathcal{P}(X)$ satisfies $\lambda_{m}(X\setminus F_{m})=0$. By Couplings on a Hilbert Space: Product Coupling, Swap, Finiteness of the Cost, Push-Forward Couplings, Modifying One Marginal, Quantisation, Gluing over a Finitely Supported Measure, the Lipschitz Bound and the Moment Bound §modification, $\pi_{12}^{m}=(\pi_{1},T_{m}\circ\pi_{2})_{\#}\pi_{12}\in\Pi(\mu,\lambda_{m})$ and $\pi_{23}^{m}=(T_{m}\circ\pi_{1},\pi_{2})_{\#}\pi_{23}\in\Pi(\lambda_{m},\nu)$. Step 1, with $\lambda'=\lambda_{m}$ and $F=F_{m}$, gives $\sigma_{m}\in\mathcal{P}(X_{(3)})$ with $(q_{1},q_{2})_{\#}\sigma_{m}=\pi_{12}^{m}$ and $(q_{2},q_{3})_{\#}\sigma_{m}=\pi_{23}^{m}$.

Step 3 (the quantised middle marginals are tight). By Couplings on a Hilbert Space: Product Coupling, Swap, Finiteness of the Cost, Push-Forward Couplings, Modifying One Marginal, Quantisation, Gluing over a Finitely Supported Measure, the Lipschitz Bound and the Moment Bound §pushforward with $S=\mathrm{id}_{X}$ and $T=T_{m}$, the coupling $(\mathrm{id}_{X},T_{m})_{\#}\lambda\in\Pi(\lambda,\lambda_{m})$ has cost $\int|y-T_{m}(y)|^{2}\lambda(dy)\le m^{-2}$, as $|y-T_{m}(y)|=|T_{m}(y)-y|$. So $\lambda_{m}\in\mathcal{P}_{2}(X)$ by the converse part of Couplings on a Hilbert Space: Product Coupling, Swap, Finiteness of the Cost, Push-Forward Couplings, Modifying One Marginal, Quantisation, Gluing over a Finitely Supported Measure, the Lipschitz Bound and the Moment Bound §cost-finite, and $W_{2}(\lambda,\lambda_{m})^{2}\le m^{-2}$ by The Quadratic Wasserstein Distance on a Hilbert Space §distance, whence $W_{2}(\lambda,\lambda_{m})\le1/m$ by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field. By The Quadratic Wasserstein Distance is a Metric on the Probability Measures with Finite Second Moment on a Hilbert Space §symmetry and The Quadratic Wasserstein Distance is a Metric on the Probability Measures with Finite Second Moment on a Hilbert Space §triangle, $W_{2}(\lambda_{m},\lambda_{\ell})\le1/m+1/\ell$; given a real $\varepsilon>0$ and $N\in\mathbb{N}$ with $2/N<\varepsilon$, this is $<\varepsilon$ for $m,\ell\ge N$, so $(\lambda_{m})$ is a Cauchy sequence in $(\mathcal{P}_{2}(X),W_{2})$ by Cauchy Sequence in a Metric Space, and it is tight in $(X,d)$ by Cauchy Sequences in the Quadratic Wasserstein Space and Weakly Convergent Sequences on a Hilbert Space are Tight §tight.

Step 4 (the gluings are tight). Fix a real $\varepsilon>0$; first $K_{1}$, then $K_{3}$ is chosen. The sets $\{\mu\}$ (by Ulam's Theorem: a Finite Borel Measure on a Complete Separable Metric Space is Tight §tight, $(X,d)$ being complete and separable by Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §space) and $\{\lambda_{m}:m\in\mathbb{N}\}$ (by Step 3 and Tight Family of Borel Measures on a Metric Space §sequence) are tight, so by Couplings on a Hilbert Space: Tightness, Closedness under Weak Convergence, and Lower Semicontinuity of the Quadratic Cost §tight and Tight Family of Borel Measures on a Metric Space §tight there is a compact $K_{1}\subseteq X\times X$ with $\pi_{12}^{m}((X\times X)\setminus K_{1})\le\varepsilon/2$ for every $m$. By Ulam's Theorem: a Finite Borel Measure on a Complete Separable Metric Space is Tight §tight choose a compact $K_{3}\subseteq X$ with $\nu(X\setminus K_{3})\le\varepsilon/2$. By A Product of Compact Subsets is Compact in the Product Metric, applied to $(X\times X,d)$ and $(X,d)$, the set $K_{1}\times K_{3}$ is compact for the topology of the product metric, which by Properties of the Product of Two Real Inner Product Spaces §metric (with $E_{1}=X\times X$, $E_{2}=X$) has the same open sets as $(X_{(3)},d)$; compactness depending only on the topology (Compact Topological Space and Compact Subset), $K_{1}\times K_{3}$ is compact in $(X_{(3)},d)$. The sets $(X\times X)\setminus K_{1}$ and $X\setminus K_{3}$ are Borel by Compact Subsets of a Metric Space are Closed and Borel §borel, and a point $w$ lies outside $K_{1}\times K_{3}$ exactly when $(q_{1},q_{2})(w)\notin K_{1}$ or $q_{3}(w)\notin K_{3}$. Since $q_{3}=\pi_{2}\circ(q_{2},q_{3})$, one has $(q_{3})_{\#}\sigma_{m}=(\pi_{2})_{\#}\pi_{23}^{m}=\nu$. By Basic Properties of a Measure §subadditivity (padding with empty sets),

$$\sigma_{m}\bigl(X_{(3)}\setminus(K_{1}\times K_{3})\bigr)\le\pi_{12}^{m}\bigl((X\times X)\setminus K_{1}\bigr)+\nu(X\setminus K_{3})\le\varepsilon$$

for every $m$, so $(\sigma_{m})$ is tight in $(X_{(3)},d)$ by Tight Family of Borel Measures on a Metric Space §sequence.

Step 5 (a weak limit). By Prokhorov's Theorem on a Metric Space: a Tight Sequence of Borel Probability Measures Has a Weakly Convergent Subsequence §subsequence on $(X_{(3)},d)$ there are a strictly increasing $(m_{i})_{i\in\mathbb{N}}$ and $\sigma\in\mathcal{P}(X_{(3)})$ with $\sigma_{m_{i}}\Rightarrow\sigma$. By Weak Convergence of Finite Borel Measures is Preserved by Continuous Maps between Metric Spaces §weak, applied to the continuous maps $(q_{1},q_{2})$ and $(q_{2},q_{3})$, we get $\pi_{12}^{m_{i}}\Rightarrow(q_{1},q_{2})_{\#}\sigma$ and $\pi_{23}^{m_{i}}\Rightarrow(q_{2},q_{3})_{\#}\sigma$ on $(X\times X,d)$.

Step 6 (identification; clause 1). Let $f:X\times X\to\mathbb{R}$ be bounded and Lipschitz with constant $L$. It is continuous by A Lipschitz Map is Uniformly Continuous and Borel by claim 3 of Borel Measurability and Bounded Integration on a Metric Space, so $f$ and $f\circ(\pi_{1},T_{m}\circ\pi_{2})$ are bounded Borel functions, the latter because $\pi_{1},\pi_{2}$ are Borel by Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §product-sigma, $T_{m}\circ\pi_{2}$ is Borel by claim 4 of Borel Measurability and Bounded Integration on a Metric Space, the pair $(\pi_{1},T_{m}\circ\pi_{2})$ is Borel by Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §pairing, and its composite with $f$ is Borel by claim 4 of Borel Measurability and Bounded Integration on a Metric Space again (likewise $f\circ(T_{m}\circ\pi_{1},\pi_{2})$ is Borel, with the roles of the coordinates exchanged); integrable against every member of $\mathcal{P}(X\times X)$ by claim 6 of Borel Measurability and Bounded Integration on a Metric Space. For $x,y\in X$, by Properties of the Product of Two Real Inner Product Spaces §metric and the inequality $2mr\le m^{2}r^{2}+1$ for real $r$ (as $(mr-1)^{2}\ge0$),

$$|f(x,T_{m}y)-f(x,y)|\le L\,d\bigl((x,T_{m}y),(x,y)\bigr)=L|T_{m}y-y|\le L\Bigl(\frac{m}{2}|T_{m}y-y|^{2}+\frac{1}{2m}\Bigr).$$

The right side, as a function of $z=(x,y)$, is $L(\frac{m}{2}h(\pi_{2}z)+\frac{1}{2m})$ with $h(y)=|T_{m}y-y|^{2}$ Borel (statement of Couplings on a Hilbert Space: Product Coupling, Swap, Finiteness of the Cost, Push-Forward Couplings, Modifying One Marginal, Quantisation, Gluing over a Finitely Supported Measure, the Lipschitz Bound and the Moment Bound). By change of variables (integrable case, then through $\pi_{2}$ with $(\pi_{2})_{\#}\pi_{12}=\lambda$), the integral theorem (claim 2 for linearity and for $|\int g|\le\int|g|$, claim 1 for monotonicity and linearity of nonnegative integrals, the two readings of the integral of the bounded nonnegative function $|g|$ agreeing by claim 6(c) of Borel Measurability and Bounded Integration on a Metric Space), the constant $\frac{L}{2m}$ having integral $\frac{L}{2m}\,\pi_{12}(X\times X)=\frac{L}{2m}$, the constant times the total mass $1$, by claim 6(a) of Borel Measurability and Bounded Integration on a Metric Space, and Step 2,

$$\Bigl|\int f\,d\pi_{12}^{m}-\int f\,d\pi_{12}\Bigr|\le\int\bigl|f\circ(\pi_{1},T_{m}\circ\pi_{2})-f\bigr|\,d\pi_{12}\le\frac{Lm}{2}\int_{X}|T_{m}y-y|^{2}\,\lambda(dy)+\frac{L}{2m}\le\frac{L}{m}.$$

As $m_{i}\ge i$, the integrals $\int f\,d\pi_{12}^{m_{i}}$ converge to $\int f\,d\pi_{12}$, so claim 1 of Portmanteau Theorem on a Metric Space on $(X\times X,d)$ gives $\pi_{12}^{m_{i}}\Rightarrow\pi_{12}$. The same argument with $f(T_{m}x,y)-f(x,y)$ and change of variables through $\pi_{1}$, where $(\pi_{1})_{\#}\pi_{23}=\lambda$, gives $\pi_{23}^{m_{i}}\Rightarrow\pi_{23}$. By claim 2 of Lipschitz Test Functions Determine a Finite Borel Measure, and Uniqueness of Weak Limits on $(X\times X,d)$ and Step 5, $(q_{1},q_{2})_{\#}\sigma=\pi_{12}$ and $(q_{2},q_{3})_{\#}\sigma=\pi_{23}$. This proves clause 1.

Step 7 (clause 2). Let $\sigma$ be a gluing and $\gamma=(q_{1},q_{3})_{\#}\sigma\in\mathcal{P}(X\times X)$. By the conventions, $(\pi_{1})_{\#}\gamma=(q_{1})_{\#}\sigma=(\pi_{1})_{\#}\pi_{12}=\mu$ and $(\pi_{2})_{\#}\gamma=(q_{3})_{\#}\sigma=(\pi_{2})_{\#}\pi_{23}=\nu$, so $\gamma\in\Pi(\mu,\nu)$ by Couplings of Two Borel Probability Measures on a Hilbert Space and Their Quadratic Cost §coupling, and the three costs are real by Couplings on a Hilbert Space: Product Coupling, Swap, Finiteness of the Cost, Push-Forward Couplings, Modifying One Marginal, Quantisation, Gluing over a Finitely Supported Measure, the Lipschitz Bound and the Moment Bound §cost-finite. With the Borel function $\varphi(z)=|\pi_{1}z-\pi_{2}z|^{2}$ of Couplings of Two Borel Probability Measures on a Hilbert Space and Their Quadratic Cost §cost, change of variables gives $I(\gamma)=\int|q_{1}-q_{3}|^{2}\,d\sigma$, $I(\pi_{12})=\int|q_{1}-q_{2}|^{2}\,d\sigma$ and $I(\pi_{23})=\int|q_{2}-q_{3}|^{2}\,d\sigma$. Pointwise $q_{1}-q_{3}=(q_{1}-q_{2})+(q_{2}-q_{3})$, so by The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §triangle, claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field and (E1), for every real $t>0$,

$$|q_{1}-q_{3}|^{2}\le(1+t)|q_{1}-q_{2}|^{2}+(1+t^{-1})|q_{2}-q_{3}|^{2}\quad\text{on }X_{(3)} .$$

Integrating by claim 1 of the integral theorem, $I(\gamma)\le(1+t)I(\pi_{12})+(1+t^{-1})I(\pi_{23})$ for every $t>0$, and (E2) gives clause 2.

Step 8 (clause 3). Let $\pi_{12}\in\Pi^{a}(\mu,\lambda)$, $\pi_{23}\in\Pi^{a}(\lambda,\nu)$, $\sigma$ a gluing and $\gamma$ as in Step 7, so $\gamma\in\Pi(\mu,\nu)$. With $D_{a}$, $n_{a}$ and $c_{a}$ of The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §borel and The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §pairs, the sets $G_{12}=(q_{1},q_{2})^{-1}(D_{a})=\{w:q_{2}w-q_{1}w\in X^{a}\}$ and $G_{23}=(q_{2},q_{3})^{-1}(D_{a})=\{w:q_{3}w-q_{2}w\in X^{a}\}$ are Borel, and $\sigma(G_{12})=\pi_{12}(D_{a})=1$ and $\sigma(G_{23})=\pi_{23}(D_{a})=1$ by Couplings of Finite Noise Cost and Their Noise Cost §finite. Let $G=G_{12}\cap G_{23}$; its complement is the union of two $\sigma$-null sets (Basic Properties of a Measure §differences), so $\sigma(X_{(3)}\setminus G)=0$ by Basic Properties of a Measure §subadditivity and $\sigma(G)=1$. For $w\in G$, $q_{3}w-q_{1}w=(q_{3}w-q_{2}w)+(q_{2}w-q_{1}w)\in X^{a}$, a linear subspace by The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §hilbert; so $G\subseteq(q_{1},q_{3})^{-1}(D_{a})$ and $\gamma(D_{a})=1$ by Basic Properties of a Measure §monotone. Moreover, for $w\in G$ and real $t>0$, by the definition of $n_{a}$, The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §triangle in the real inner product space $(X^{a},\langle\cdot,\cdot\rangle_{a})$ of The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §hilbert, claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field and (E1),

$$c_{a}\bigl((q_{1},q_{3})w\bigr)=|q_{3}w-q_{1}w|_{a}^{2}\le(1+t)\,c_{a}\bigl((q_{1},q_{2})w\bigr)+(1+t^{-1})\,c_{a}\bigl((q_{2},q_{3})w\bigr),$$

where $c_{a}((q_{i},q_{j})w)=n_{a}(q_{j}w-q_{i}w)$. All three functions are nonnegative and Borel, as composites of Borel maps (claim 4 of Borel Measurability and Bounded Integration on a Metric Space), and the inequality holds $\sigma$-almost everywhere. By The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §comparison, claim 1 of the integral theorem and change of variables,

$$\int_{X\times X}c_{a}\,d\gamma\le(1+t)\int c_{a}\,d\pi_{12}+(1+t^{-1})\int c_{a}\,d\pi_{23}=(1+t)I^{a}(\pi_{12})+(1+t^{-1})I^{a}(\pi_{23})<\infty,$$

using Couplings of Finite Noise Cost and Their Noise Cost §cost. Hence $\gamma$ has finite noise cost by Couplings of Finite Noise Cost and Their Noise Cost §finite, $\gamma\in\Pi^{a}(\mu,\nu)$ by Couplings of Finite Noise Cost and Their Noise Cost §couplings, and $I^{a}(\gamma)\le(1+t)I^{a}(\pi_{12})+(1+t^{-1})I^{a}(\pi_{23})$ for every $t>0$; (E2) gives clause 3.