Proof of The Displacement Pairing of a Square-Integrable Vector Field Along a 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. Throughout, $\mathrm{pr}_{1}$ and $\mathrm{pr}_{2}$ are Borel by Probability Measures on Euclidean Space and Random Vectors: Standing Notation (v), and $(\mathrm{pr}_{1})_{\#}\pi=\mu$, $(\mathrm{pr}_{2})_{\#}\pi=\nu$ by Couplings of Two Probability Measures on Euclidean Space and Their Quadratic Cost (i).

Claim 1. Fix a representative of $\eta$, again written $\eta$: a Borel map $\mathbb{R}^{d}\to\mathbb{R}^{d}$ with $\int_{\mathbb{R}^{d}}\lVert\eta\rVert^{2}\,d\mu<\infty$, as Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation (ii) describes the elements of $L^{2}(\mu;\mathbb{R}^{d})$.

The composite $\eta\circ\mathrm{pr}_{1}:\mathbb{R}^{d+d}\to\mathbb{R}^{d}$ is Borel by Probability Measures on Euclidean Space and Random Vectors: Standing Notation (ii). Its squared norm 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 (iv) and Nonnegativity of Squares in an Ordered Field, and by the change-of-variables formula of Probability Measures on Euclidean Space and Random Vectors: Standing Notation (iv) applied to $\mathrm{pr}_{1}$,

$$\int_{\mathbb{R}^{d+d}}\lVert\eta\circ\mathrm{pr}_{1}\rVert^{2}\,d\pi=\int_{\mathbb{R}^{d}}\lVert\eta\rVert^{2}\,d\mu=\lVert\eta\rVert_{\mu}^{2}<\infty .$$

Hence the class of $\eta\circ\mathrm{pr}_{1}$ belongs to $L^{2}(\pi;\mathbb{R}^{d})$ and has norm $\lVert\eta\rVert_{\mu}$ there, by Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation (ii). By The Coordinate Fields of a Coupling, and the Second Moment as a Lipschitz Function of the Wasserstein Distance (i), read with $d$ in place of the dimension $m$ there, the classes of $\mathrm{pr}_{1}$ and $\mathrm{pr}_{2}$ belong to $L^{2}(\pi;\mathbb{R}^{d})$ with

$$\lVert\mathrm{pr}_{1}-\mathrm{pr}_{2}\rVert_{\pi}^{2}=I(\pi),$$

so $\lVert\mathrm{pr}_{1}-\mathrm{pr}_{2}\rVert_{\pi}=\sqrt{I(\pi)}$ by Existence and Uniqueness of the Nonnegative Square Root, the norm being nonnegative.

Both $\eta\circ\mathrm{pr}_{1}$ and $\mathrm{pr}_{1}-\mathrm{pr}_{2}$ are thus elements of $L^{2}(\pi;\mathbb{R}^{d})$, so by Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation (ii), which fixes the inner product of that space as the integral of the pointwise dot product of representatives, the function $z\mapsto\eta(x)\cdot(x-y)$ is integrable with respect to $\pi$ and

$$\int_{\mathbb{R}^{d+d}}\eta(x)\cdot(x-y)\,\pi(dz)=\bigl\langle\eta\circ\mathrm{pr}_{1},\ \mathrm{pr}_{1}-\mathrm{pr}_{2}\bigr\rangle_{\pi}.$$

The function $z\mapsto\eta(x)\cdot(y-x)$ is $(-1)$ times this integrand, by claim 2 of Zero Products and Elementary Identities in a Field together with the bilinearity of the dot product, The Dot Product on $\mathbb{R}^n$ Is Symmetric, Bilinear and Positive Definite; it is therefore Borel and integrable with respect to $\pi$, with

$$\mathcal{J}(\eta,\pi)=-\bigl\langle\eta\circ\mathrm{pr}_{1},\ \mathrm{pr}_{1}-\mathrm{pr}_{2}\bigr\rangle_{\pi},$$

by claim 2 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion applied with the scalars $-1$ and $0$.

It remains to see that this number does not depend on the representative. Let $\eta'$ be another representative of the same class, so that $\mu(\{\eta=\eta'\})=1$ by Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation (ii). The map $\eta-\eta'$ is Borel, its components being differences of Borel real functions by claim 2 of Arithmetic of Measurable Real- and [0,∞]-Valued Maps and a map into $\mathbb{R}^{d}$ with Borel components being Borel by Probability Measures on Euclidean Space and Random Vectors: Standing Notation (ii); the set $\{\eta=\eta'\}$ is its preimage of the one-point set $\{0_{\mathbb{R}^{d}}\}$, which is Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (v), and is therefore Borel. Hence $N=\mathbb{R}^{d}\setminus\{\eta=\eta'\}$ is Borel, being the complement of a Borel set, and $\mu(N)=0$ by claim 3 of Basic Properties of a Measure, $\mu$ being finite with $\mu(\mathbb{R}^{d})=1$. Since $(\mathrm{pr}_{1})_{\#}\pi=\mu$, the Borel set $\mathrm{pr}_{1}^{-1}(N)$ satisfies $\pi(\mathrm{pr}_{1}^{-1}(N))=\mu(N)=0$ by Probability Measures on Euclidean Space and Random Vectors: Standing Notation (iv). Off $\mathrm{pr}_{1}^{-1}(N)$ the two integrands $z\mapsto\eta(x)\cdot(y-x)$ and $z\mapsto\eta'(x)\cdot(y-x)$ agree, so they agree $\pi$-almost everywhere and their integrals coincide by The Lebesgue Integral and Null Sets (iii).

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

$$\bigl|\mathcal{J}(\eta,\pi)\bigr|=\bigl|\bigl\langle\eta\circ\mathrm{pr}_{1},\ \mathrm{pr}_{1}-\mathrm{pr}_{2}\bigr\rangle_{\pi}\bigr|\le\lVert\eta\circ\mathrm{pr}_{1}\rVert_{\pi}\,\lVert\mathrm{pr}_{1}-\mathrm{pr}_{2}\rVert_{\pi}=\lVert\eta\rVert_{\mu}\,\sqrt{I(\pi)},$$

the first equality using the symmetry of the absolute value, claim 2 of Properties of the Absolute Value in an Ordered Field, and the last the two norm computations of claim 1.

Claim 3. If $\eta$ and $\xi$ are represented by Borel maps and $a,b\in\mathbb{R}$, then $a\eta+b\xi$ is represented by the pointwise combination $x\mapsto a\,\eta(x)+b\,\xi(x)$, by Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation (ii), and the value $(a\,\eta(x)+b\,\xi(x))\cdot(y-x)$ equals $a\,(\eta(x)\cdot(y-x))+b\,(\xi(x)\cdot(y-x))$ for every $z$, by the bilinearity of the dot product, The Dot Product on $\mathbb{R}^n$ Is Symmetric, Bilinear and Positive Definite. Both $z\mapsto\eta(x)\cdot(y-x)$ and $z\mapsto\xi(x)\cdot(y-x)$ are $\pi$-integrable by claim 1, so claim 2 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion gives the asserted identity.

Claim 4. By hypothesis $S-\mathrm{id}$ is a Borel map with $\int_{\mathbb{R}^{d}}\lVert S-\mathrm{id}\rVert^{2}\,d\mu<\infty$, so its class belongs to $L^{2}(\mu;\mathbb{R}^{d})$ by Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation (ii), with $\lVert S-\mathrm{id}\rVert_{\mu}^{2}=\int_{\mathbb{R}^{d}}\lVert S-\mathrm{id}\rVert^{2}\,d\mu$. Since $\mathrm{id}$ belongs to $L^{2}(\mu;\mathbb{R}^{d})$ by Basic Properties of the Tangent Space: Closed Subspace, the Identity Map Belongs to It, Second-Moment Limits, and Representation of Bounded Functionals on Gradients (ii) and that space is a real vector space by Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation (ii), the class of $S=(S-\mathrm{id})+\mathrm{id}$ belongs to it too, so $\int_{\mathbb{R}^{d}}\lVert S\rVert^{2}\,d\mu<\infty$ and $S_{\#}\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ 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 (iv), applied with $\mathrm{id}$ and $S$ in the roles of the two Borel maps, $\pi_{S}=(\mathrm{id},S)_{\#}\mu$ belongs to $\Pi(\mathrm{id}_{\#}\mu,S_{\#}\mu)=\Pi(\mu,S_{\#}\mu)$, the push-forward of $\mu$ by the identity being $\mu$ by Probability Measures on Euclidean Space and Random Vectors: Standing Notation (iv), and

$$I(\pi_{S})=\int_{\mathbb{R}^{d}}\lVert x-S(x)\rVert^{2}\,\mu(dx)=\lVert S-\mathrm{id}\rVert_{\mu}^{2},$$

the second equality because $\lVert x-S(x)\rVert=\lVert S(x)-x\rVert$ for every $x$, by the absolute homogeneity of the Euclidean norm, 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, applied with the scalar $-1$.

For the pairing, the change-of-variables formula of Probability Measures on Euclidean Space and Random Vectors: Standing Notation (iv) applied to the Borel map $(\mathrm{id},S)$ and to the $\pi_{S}$-integrable function $z\mapsto\eta(x)\cdot(y-x)$ of claim 1 gives, using $\mathrm{pr}_{1}((\mathrm{id},S)(x))=x$ and $\mathrm{pr}_{2}((\mathrm{id},S)(x))=S(x)$ from Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (i),

$$\mathcal{J}(\eta,\pi_{S})=\int_{\mathbb{R}^{d}}\eta(x)\cdot\bigl(S(x)-x\bigr)\,\mu(dx)=\langle\eta,\,S-\mathrm{id}\rangle_{\mu},$$

the last equality by Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation (ii), both classes lying in $L^{2}(\mu;\mathbb{R}^{d})$.

Claim 5. The constant map with value $c$ is Borel, being continuous. For any probability measure $\lambda$ on a Euclidean space, the constant function with value $\lVert c\rVert^{2}$ is $\lVert c\rVert^{2}\,\mathbf{1}$ on that space, so its integral against $\lambda$ is $\lVert c\rVert^{2}$ by claim 1 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion for the nonnegative scalar $\lVert c\rVert^{2}$, by The Integral of an Indicator Function is the Measure of the Set and by Probability Measures on Euclidean Space and Random Vectors: Standing Notation (iii). Applying this with $\lambda=\mu$ shows that the class $\eta$ of the constant map lies in $L^{2}(\mu;\mathbb{R}^{d})$, and applying it with $\lambda=\pi$ shows that the class of the constant map on $\mathbb{R}^{d+d}$ with value $c$ lies in $L^{2}(\pi;\mathbb{R}^{d})$.

By claim 1 the function $z\mapsto c\cdot(y-x)$ is $\pi$-integrable, and by the bilinearity of the dot product, The Dot Product on $\mathbb{R}^n$ Is Symmetric, Bilinear and Positive Definite, it equals $c\cdot y-c\cdot x$ pointwise. The functions $z\mapsto c\cdot\mathrm{pr}_{2}(z)$ and $z\mapsto c\cdot\mathrm{pr}_{1}(z)$ are $\pi$-integrable, being the pointwise dot products of that constant class of $L^{2}(\pi;\mathbb{R}^{d})$ with the classes of $\mathrm{pr}_{2}$ and of $\mathrm{pr}_{1}$, which lie in $L^{2}(\pi;\mathbb{R}^{d})$ by The Coordinate Fields of a Coupling, and the Second Moment as a Lipschitz Function of the Wasserstein Distance (i), the integrability of a pointwise dot product of two elements of that space being recorded in Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation (ii); so claim 2 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion gives

$$\mathcal{J}(\eta,\pi)=\int_{\mathbb{R}^{d+d}}c\cdot\mathrm{pr}_{2}\,d\pi-\int_{\mathbb{R}^{d+d}}c\cdot\mathrm{pr}_{1}\,d\pi .$$

By the change-of-variables formula of Probability Measures on Euclidean Space and Random Vectors: Standing Notation (iv) applied to $\mathrm{pr}_{2}$ and to $\mathrm{pr}_{1}$, these two integrals are $\int_{\mathbb{R}^{d}}c\cdot y\,\nu(dy)$ and $\int_{\mathbb{R}^{d}}c\cdot x\,\mu(dx)$. Writing $c\cdot x=\sum_{i=1}^{d}c_{i}x_{i}$ by Difference, Dot Product, and Orthogonality in $\mathbb{R}^n$ and using claim 2 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion together with claim 1 of The Mean of a Square-Integrable Probability Measure, Its Lift, Its Centring, and Functions of the Mean and Centred Integrals as Test Functions, which supplies the integrability of each coordinate map and the identity $m(\mu)_{i}=\int_{\mathbb{R}^{d}}x_{i}\,\mu(dx)$, the second integral equals $\sum_{i=1}^{d}c_{i}\,m(\mu)_{i}=c\cdot m(\mu)$, and likewise the first equals $c\cdot m(\nu)$. Subtracting and using The Dot Product on $\mathbb{R}^n$ Is Symmetric, Bilinear and Positive Definite once more gives $\mathcal{J}(\eta,\pi)=c\cdot(m(\nu)-m(\mu))$.