Proof of A Triangle Inequality for Discrepancies Along a Composite Coupling

Each result cited is universally quantified over the data in its own statement. Integrals against push-forwards are transported by the change-of-variables formula, called "change of variables" below. $L^{2}(\sigma;\mathbb{R}^{d})$ is the space of square-integrable vector fields against $\sigma\in\mathcal{P}(\mathbb{R}^{3d})$, a real Hilbert space by that clause, with norm $\lVert\cdot\rVert_{\sigma}$; the class of a Borel map square-integrable against $\sigma$ is written with the same symbol as the map.

Step 1 (Marginals of the gluing). By the definition of a gluing (Gluing Two Couplings over a Common Middle Marginal, and the Composite Coupling §glued), $(\mathrm{q}_{1},\mathrm{q}_{2})_{\#}\sigma=\pi_{12}$ and $(\mathrm{q}_{2},\mathrm{q}_{3})_{\#}\sigma=\pi_{23}$. For a Borel set $B\subseteq\mathbb{R}^{d}$ one has $\mathrm{q}_{1}^{-1}(B)=(\mathrm{q}_{1},\mathrm{q}_{2})^{-1}(\mathrm{pr}_{1}^{-1}(B))$, $\mathrm{q}_{2}^{-1}(B)=(\mathrm{q}_{1},\mathrm{q}_{2})^{-1}(\mathrm{pr}_{2}^{-1}(B))$ and $\mathrm{q}_{3}^{-1}(B)=(\mathrm{q}_{2},\mathrm{q}_{3})^{-1}(\mathrm{pr}_{2}^{-1}(B))$; as $\pi_{12}\in\Pi(\nu,\rho)$ and $\pi_{23}\in\Pi(\rho,\mu)$, this gives $(\mathrm{q}_{1})_{\#}\sigma=\nu$, $(\mathrm{q}_{2})_{\#}\sigma=\rho$ and $(\mathrm{q}_{3})_{\#}\sigma=\mu$.

Step 2 (The three fields over the gluing). Fix representatives of $q$, $\eta$ and $\theta$, that is, Borel maps $\mathbb{R}^{d}\to\mathbb{R}^{d}$ with finite square integrals against $\nu$, $\rho$ and $\mu$ respectively (Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation §fields). The maps $q\circ\mathrm{q}_{1}$, $\eta\circ\mathrm{q}_{2}$ and $\theta\circ\mathrm{q}_{3}$ from $\mathbb{R}^{3d}$ to $\mathbb{R}^{d}$ are Borel as compositions of Borel maps, and by change of variables and Step 1,

and likewise for $\eta\circ\mathrm{q}_{2}$ against $\rho$ and $\theta\circ\mathrm{q}_{3}$ against $\mu$. So their classes $\hat{q}$, $\hat{\eta}$, $\hat{\theta}$ lie in $L^{2}(\sigma;\mathbb{R}^{d})$.

Step 3 (Discrepancies as distances). The function $z\mapsto\lVert q(x)-\eta(y)\rVert^{2}$ on $\mathbb{R}^{d+d}$ 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 §functions, applied to the Borel maps $q\circ\mathrm{pr}_{1}$ and $\eta\circ\mathrm{pr}_{2}$, and its composition with $(\mathrm{q}_{1},\mathrm{q}_{2})$ is $\lVert q\circ\mathrm{q}_{1}-\eta\circ\mathrm{q}_{2}\rVert^{2}$. By change of variables through $(\mathrm{q}_{1},\mathrm{q}_{2})$ and the definition of $\lVert\cdot\rVert_{\sigma}$,

In the same way, through $(\mathrm{q}_{2},\mathrm{q}_{3})$ and through $(\mathrm{q}_{1},\mathrm{q}_{3})$, with $(\mathrm{q}_{1},\mathrm{q}_{3})_{\#}\sigma=\pi_{13}$,

These three integrals are the discrepancies of the statement, which do not depend on the representatives chosen by The Discrepancy of Two Square-Integrable Vector Fields Along a Coupling of Their Base Measures §well-defined.

Step 4 (Conclusion). In $L^{2}(\sigma;\mathbb{R}^{d})$, $\hat{q}-\hat{\theta}=(\hat{q}-\hat{\eta})+(\hat{\eta}-\hat{\theta})$, so the triangle inequality The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §triangle gives $\lVert\hat{q}-\hat{\theta}\rVert_{\sigma}\le\lVert\hat{q}-\hat{\eta}\rVert_{\sigma}+\lVert\hat{\eta}-\hat{\theta}\rVert_{\sigma}$. Each norm is the nonnegative square root of the corresponding integral of Step 3, which is the claimed inequality.