Proof of The Cross Pairing of Two Square-Integrable Vector Fields Along a Coupling

Each result cited is universally quantified over the data in its own statement. By Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation (ii), in force through The Intrinsic Calculus on the Wasserstein Space: Standing Notation, a representative of $q$ is a Borel map $\mathbb{R}^{d}\to\mathbb{R}^{d}$ whose squared norm has finite integral against $\nu$, and likewise for $\eta$ and $\mu$. Fix representatives, again written $q$ and $\eta$. Since $\pi\in\Pi(\nu,\mu)$, the push-forwards of $\pi$ under $\mathrm{pr}_{1}$ and $\mathrm{pr}_{2}$ are $\nu$ and $\mu$ (Couplings of Two Probability Measures on Euclidean Space and Their Quadratic Cost (i)). Put

$$A=q\circ\mathrm{pr}_{1},\qquad B=\eta\circ\mathrm{pr}_{2},$$

maps $\mathbb{R}^{d+d}\to\mathbb{R}^{d}$; they are Borel as compositions of Borel maps (Probability Measures on Euclidean Space and Random Vectors: Standing Notation (ii)), the projections being Borel by Probability Measures on Euclidean Space and Random Vectors: Standing Notation (v). By construction $A(z)\cdot B(z)=q(x)\cdot\eta(y)$ for every $z\in\mathbb{R}^{d+d}$.

Step 1 (square integrals). The change-of-variables formula of Probability Measures on Euclidean Space and Random Vectors: Standing Notation (iv), applied to the nonnegative Borel function $w\mapsto\lVert q(w)\rVert^{2}$ (Borel as the composition of the Borel map $q$ with the map $x\mapsto\lVert x\rVert^{2}$, which is Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (iv), compositions of Borel maps being Borel by Probability Measures on Euclidean Space and Random Vectors: Standing Notation (ii)) and the map $\mathrm{pr}_{1}$, gives

$$\int_{\mathbb{R}^{d+d}}\lVert A\rVert^{2}\,d\pi=\int_{\mathbb{R}^{d}}\lVert q\rVert^{2}\,d\bigl((\mathrm{pr}_{1})_{\#}\pi\bigr)=\int_{\mathbb{R}^{d}}\lVert q\rVert^{2}\,d\nu=\lVert q\rVert_{\nu}^{2}<\infty,$$

and in the same way, with $\mathrm{pr}_{2}$, $\int_{\mathbb{R}^{d+d}}\lVert B\rVert^{2}\,d\pi=\lVert\eta\rVert_{\mu}^{2}<\infty$. Hence, by Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation (ii) read with $d+d$ as the dimension of the base space and $d$ as the dimension of the values, the classes of $A$ and $B$ belong to the real Hilbert space $L^{2}(\pi;\mathbb{R}^{d})$, with

$$\lVert A\rVert_{\pi}=\lVert q\rVert_{\nu},\qquad\lVert B\rVert_{\pi}=\lVert\eta\rVert_{\mu}. \tag{1}$$

Step 2 (claim 1). The function $z\mapsto A(z)\cdot B(z)$ is Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (iv), which also gives $|A(z)\cdot B(z)|\le\tfrac12\bigl(\lVert A(z)\rVert^{2}+\lVert B(z)\rVert^{2}\bigr)$ for every $z$. The right side has finite integral by Step 1 and claim 1 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion, so $|A\cdot B|$ has finite integral by the monotonicity in the same claim; that is, $A\cdot B$ is integrable with respect to $\pi$. By the definition of the inner product of $L^{2}(\pi;\mathbb{R}^{d})$ in Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation (ii),

$$\int_{\mathbb{R}^{d+d}}A\cdot B\,d\pi=\langle A,B\rangle_{\pi}. \tag{2}$$

Independence of the representatives: let $q'$ be another representative of $q$. By the relation $\sim_{\nu}$ of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation (ii) the Borel set $N=\{w\in\mathbb{R}^{d}:q(w)\neq q'(w)\}$, the complement of the set $\{q=q'\}$ of full $\nu$-measure, satisfies $\nu(N)=0$. The set of $z$ with $q(\mathrm{pr}_{1}(z))\neq q'(\mathrm{pr}_{1}(z))$ is $\mathrm{pr}_{1}^{-1}(N)$, whose $\pi$-measure is $\bigl((\mathrm{pr}_{1})_{\#}\pi\bigr)(N)=\nu(N)=0$ by the definition of the push-forward in Probability Measures on Euclidean Space and Random Vectors: Standing Notation (iv). So the Borel functions $A\cdot B$ and $(q'\circ\mathrm{pr}_{1})\cdot B$ agree $\pi$-almost everywhere, and by The Lebesgue Integral and Null Sets (iii) the second is integrable with the same integral. The same argument with $\mathrm{pr}_{2}$ and $\mu$ handles a change of the representative of $\eta$, and a change of both is two successive changes. This proves claim 1, and by (2)

$$\mathcal{K}(q,\eta,\pi)=\langle A,B\rangle_{\pi}. \tag{3}$$

Step 3 (claim 2). By (3), The Cauchy–Schwarz Inequality in a Real Inner Product Space in the real inner product space $L^{2}(\pi;\mathbb{R}^{d})$, and (1),

$$\bigl|\mathcal{K}(q,\eta,\pi)\bigr|=\bigl|\langle A,B\rangle_{\pi}\bigr|\le\lVert A\rVert_{\pi}\lVert B\rVert_{\pi}=\lVert q\rVert_{\nu}\lVert\eta\rVert_{\mu}.$$

Step 4 (claim 3). Let $q'\in L^{2}(\nu;\mathbb{R}^{d})$ and $a,b\in\mathbb{R}$, and fix a representative of $q'$, again written $q'$. By Square-Integrable Random Vectors: Coordinates, Operations, Almost Sure Equality and the Mean-Square Form (v), applied as in Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation (ii) to the probability space $(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}),\nu)$, the class $a\,q+b\,q'$ has the pointwise combination $w\mapsto a\,q(w)+b\,q'(w)$ as a representative; by claim 1 we may compute $\mathcal{K}(a\,q+b\,q',\eta,\pi)$ with it. Writing $A'=q'\circ\mathrm{pr}_{1}$, for every $z$

$$\bigl(a\,A(z)+b\,A'(z)\bigr)\cdot B(z)=a\,A(z)\cdot B(z)+b\,A'(z)\cdot B(z)$$

by The Dot Product on $\mathbb{R}^n$ Is Symmetric, Bilinear and Positive Definite. Both functions on the right are integrable by Step 2 (applied to $q$ and to $q'$), so claim 2 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion gives $\mathcal{K}(a\,q+b\,q',\eta,\pi)=a\,\mathcal{K}(q,\eta,\pi)+b\,\mathcal{K}(q',\eta,\pi)$. Linearity in the second slot is the same argument with $\eta,\eta'$, the projection $\mathrm{pr}_{2}$ and the probability space $(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}),\mu)$.

Step 5 (claim 4). For every $z$, claim 1 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square ($\lVert w\rVert^{2}=w\cdot w$) together with the bilinearity and symmetry of the dot product (The Dot Product on $\mathbb{R}^n$ Is Symmetric, Bilinear and Positive Definite) gives

$$\lVert A(z)-B(z)\rVert^{2}=\lVert A(z)\rVert^{2}-2\,A(z)\cdot B(z)+\lVert B(z)\rVert^{2}.$$

The three functions on the right are integrable by Steps 1 and 2. The function on the left is nonnegative and Borel, and its integral is the discrepancy of $q$ and $\eta$ along $\pi$, a real number by The Discrepancy of Two Square-Integrable Vector Fields Along a Coupling of Their Base Measures (i), computed there with any representatives; so it is integrable. Integrating with claim 2 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion and using Step 1 and claim 1 yields claim 4.

Step 6 (claim 5). Let $\mu=\nu$ and $\pi=(\mathrm{id},\mathrm{id})_{\#}\nu$, which belongs to $\Pi(\nu,\nu)$ 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 (iv). By Step 2 the Borel function $A\cdot B$ is integrable with respect to $\pi$, so the change-of-variables formula of Probability Measures on Euclidean Space and Random Vectors: Standing Notation (iv), applied to it and to the map $(\mathrm{id},\mathrm{id})$, which is Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (ii), $\mathrm{id}$ being continuous and hence Borel, gives

$$\int_{\mathbb{R}^{d+d}}A\cdot B\,d\pi=\int_{\mathbb{R}^{d}}(A\cdot B)\circ(\mathrm{id},\mathrm{id})\,d\nu .$$

For $w\in\mathbb{R}^{d}$ the point $(\mathrm{id},\mathrm{id})(w)$ is $\iota(w,w)$ by the definition of the pairing in Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (ii), so both of its projections equal $w$ by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (i), and $(A\cdot B)\bigl((\mathrm{id},\mathrm{id})(w)\bigr)=q(w)\cdot\eta(w)$. By claim 1 and the definition of the inner product of $L^{2}(\nu;\mathbb{R}^{d})$ in Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation (ii), $\mathcal{K}(q,\eta,\pi)=\int_{\mathbb{R}^{d}}q\cdot\eta\,d\nu=\langle q,\eta\rangle_{\nu}$. $\blacksquare$