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

In the setting of Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation, let $\mu,\lambda,\nu\in\mathcal{P}_{2}(X)$, the set of The Second Moment of a Borel Probability Measure on a Hilbert Space and the Probability Measures with Finite Second Moment §space, and let $\pi_{12}\in\Pi(\mu,\lambda)$ and $\pi_{23}\in\Pi(\lambda,\nu)$ be couplings. The sets $\Pi^{a}(\cdot,\cdot)$ of couplings of finite noise cost and the noise cost $I^{a}$ are those of Couplings of Finite Noise Cost and Their Noise Cost, and square roots are the nonnegative square roots of Existence and Uniqueness of the Nonnegative Square Root.

Let $X_{(3)}$ denote the product $(X\times X)\times X$ of the real Hilbert spaces $X\times X$ and $X$, which is a real Hilbert space by Properties of the Product of Two Real Inner Product Spaces §hilbert, with the norm $|\cdot|$ and distance $d$ of Properties of the Product of Two Real Inner Product Spaces §inner-product-space. For $w=((u,v),s)\in X_{(3)}$ let $q_{1}(w)=u$, $q_{2}(w)=v$ and $q_{3}(w)=s$. Each $q_{i}:X_{(3)}\to X$ is linear with $|q_{i}(w)|\le|w|$, by Properties of the Product of Two Real Inner Product Spaces §coordinates applied to both products, hence Lipschitz with constant $1$ and continuous, and so Borel by claim 3 of Borel Measurability and Bounded Integration on a Metric Space; the pairings $(q_{1},q_{2})$, $(q_{2},q_{3})$ and $(q_{1},q_{3})$ are therefore Borel maps from $X_{(3)}$ to $X\times X$ by Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §pairing. $\mathcal{B}(X_{(3)})$ is the Borel $\sigma$-algebra of $(X_{(3)},d)$, $\mathcal{P}(X_{(3)})$ is the set of Borel measures $\sigma$ on $(X_{(3)},d)$ with $\sigma(X_{(3)})=1$, and push-forwards by Borel maps defined on $X_{(3)}$ are the image measures of claim 1 of that lemma, as in Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §pushforward. Then the following hold.

1. (Gluing)¶ There is $\sigma\in\mathcal{P}(X_{(3)})$ with

$$(q_{1},q_{2})_{\#}\sigma=\pi_{12},\qquad(q_{2},q_{3})_{\#}\sigma=\pi_{23};$$

such a $\sigma$ is called a gluing of $\pi_{12}$ and $\pi_{23}$.

2. (The composite coupling)¶ Let $\sigma$ be a gluing of $\pi_{12}$ and $\pi_{23}$. Then $(q_{1},q_{3})_{\#}\sigma\in\Pi(\mu,\nu)$, the quadratic costs $I(\pi_{12})$, $I(\pi_{23})$ and $I((q_{1},q_{3})_{\#}\sigma)$ are real numbers 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, and

$$\sqrt{I\bigl((q_{1},q_{3})_{\#}\sigma\bigr)}\le\sqrt{I(\pi_{12})}+\sqrt{I(\pi_{23})}.$$

3. (The triangle inequality for the noise cost)¶ Suppose that $\pi_{12}\in\Pi^{a}(\mu,\lambda)$ and $\pi_{23}\in\Pi^{a}(\lambda,\nu)$, and let $\sigma$ be a gluing of $\pi_{12}$ and $\pi_{23}$. Then $(q_{1},q_{3})_{\#}\sigma\in\Pi^{a}(\mu,\nu)$ and

$$\sqrt{I^{a}\bigl((q_{1},q_{3})_{\#}\sigma\bigr)}\le\sqrt{I^{a}(\pi_{12})}+\sqrt{I^{a}(\pi_{23})}.$$