Proof of The Discrepancy of Two Square-Integrable Vector Fields Along a Coupling of Their Base Measures

Each result cited is universally quantified over the data in its own statement. Throughout, $q$ and $\eta$ also denote fixed representatives: Borel maps $\mathbb{R}^{d}\to\mathbb{R}^{d}$ with $\int_{\mathbb{R}^{d}}\lVert q\rVert^{2}\,d\nu<\infty$ and $\int_{\mathbb{R}^{d}}\lVert\eta\rVert^{2}\,d\mu<\infty$, and $\lVert q\rVert_{\nu}^{2}=\int_{\mathbb{R}^{d}}\lVert q\rVert^{2}\,d\nu$, $\lVert\eta\rVert_{\mu}^{2}=\int_{\mathbb{R}^{d}}\lVert\eta\rVert^{2}\,d\mu$, by Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation (ii) and the defining property of the nonnegative square root (Existence and Uniqueness of the Nonnegative Square Root). By Couplings of Two Probability Measures on Euclidean Space and Their Quadratic Cost (i), $\pi\in\mathcal{P}(\mathbb{R}^{d+d})$ satisfies $(\mathrm{pr}_{1})_{\#}\pi=\nu$ and $(\mathrm{pr}_{2})_{\#}\pi=\mu$, that is, $\pi(\mathrm{pr}_{1}^{-1}(A))=\nu(A)$ and $\pi(\mathrm{pr}_{2}^{-1}(B))=\mu(B)$ for all $A,B\in\mathcal{B}(\mathbb{R}^{d})$.

Step 1: measurability and nonnegativity. The projections are Borel by Probability Measures on Euclidean Space and Random Vectors: Standing Notation (v), so the compositions $q\circ\mathrm{pr}_{1}$ and $\eta\circ\mathrm{pr}_{2}$ are Borel maps $\mathbb{R}^{d+d}\to\mathbb{R}^{d}$ by Probability Measures on Euclidean Space and Random Vectors: Standing Notation (ii). Hence $f_{q,\eta}$, which is the map $z\mapsto\lVert(q\circ\mathrm{pr}_{1})(z)-(\eta\circ\mathrm{pr}_{2})(z)\rVert^{2}$, is Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (iv), applied on the measurable space $(\mathbb{R}^{d+d},\mathcal{B}(\mathbb{R}^{d+d}))$; and it is nonnegative by claim 2 of Nonnegativity of Squares in an Ordered Field. Likewise $z\mapsto\lVert q(x)\rVert^{2}$ and $z\mapsto\lVert\eta(y)\rVert^{2}$ are nonnegative Borel functions on $\mathbb{R}^{d+d}$, being the compositions of the Borel maps $x\mapsto\lVert q(x)\rVert^{2}$ and $y\mapsto\lVert\eta(y)\rVert^{2}$ (Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (iv)) with the projections.

Step 2: the bound. For every $z\in\mathbb{R}^{d+d}$, the inequality $\lVert a-b\rVert^{2}\le2\lVert a\rVert^{2}+2\lVert b\rVert^{2}$ of Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (iv) with $a=q(x)$ and $b=\eta(y)$ gives $f_{q,\eta}(z)\le2\lVert q(x)\rVert^{2}+2\lVert\eta(y)\rVert^{2}$. By the monotonicity, additivity and homogeneity of the integral of nonnegative functions (claim 1 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion),

$$\int_{\mathbb{R}^{d+d}}f_{q,\eta}\,d\pi\le2\int_{\mathbb{R}^{d+d}}\lVert q(x)\rVert^{2}\,\pi(dz)+2\int_{\mathbb{R}^{d+d}}\lVert\eta(y)\rVert^{2}\,\pi(dz).$$

By the change-of-variables formula of Probability Measures on Euclidean Space and Random Vectors: Standing Notation (iv), applied to the Borel map $\mathrm{pr}_{1}$ and the nonnegative Borel function $x\mapsto\lVert q(x)\rVert^{2}$,

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

and in the same way, with $\mathrm{pr}_{2}$ and $\mu$, $\int_{\mathbb{R}^{d+d}}\lVert\eta(y)\rVert^{2}\,\pi(dz)=\lVert\eta\rVert_{\mu}^{2}$. This proves the displayed bound of claim 1, and the right-hand side is a real number.

Step 3: independence of the representatives. Let $q'$ and $\eta'$ be other representatives of the same classes, so that $\nu(\{q=q'\})=1$ and $\mu(\{\eta=\eta'\})=1$ by Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation (ii), where $\{q=q'\}\in\mathcal{B}(\mathbb{R}^{d})$ and $\{\eta=\eta'\}\in\mathcal{B}(\mathbb{R}^{d})$ by Random Vectors: Coordinates, Functions, Pairs and Almost-Sure Equality (iv), applied to the random vectors $q,q'$ on the probability space $(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}),\nu)$ of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation (i), respectively to $\eta,\eta'$ on $(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}),\mu)$. Put $A=\mathbb{R}^{d}\setminus\{q=q'\}$ and $B=\mathbb{R}^{d}\setminus\{\eta=\eta'\}$, Borel sets with $\nu(A)=\nu(\mathbb{R}^{d})-\nu(\{q=q'\})=1-1=0$ and likewise $\mu(B)=0$, by claim 3 of Basic Properties of a Measure, the measures being finite. Then $\pi(\mathrm{pr}_{1}^{-1}(A))=\nu(A)=0$ and $\pi(\mathrm{pr}_{2}^{-1}(B))=\mu(B)=0$. For $z\notin\mathrm{pr}_{1}^{-1}(A)$ one has $q(x)=q'(x)$, and for $z\notin\mathrm{pr}_{2}^{-1}(B)$ one has $\eta(y)=\eta'(y)$; so each of the two equalities holds for $\pi$-almost every $z$, hence both hold simultaneously for $\pi$-almost every $z$ by The Lebesgue Integral and Null Sets (i), and at every such $z$ one has $f_{q,\eta}(z)=f_{q',\eta'}(z)$. Both functions being nonnegative and Borel by Step 1, applied also to the representatives $q'$ and $\eta'$, their integrals against $\pi$ agree by the equality case of The Lebesgue Integral and Null Sets (iii). This completes the proof of claim 1.

Step 4: claim 2. Let $\mu=\nu$ and $\pi=(\mathrm{id},\mathrm{id})_{\#}\nu$, which lies in $\Pi(\nu,\nu)$ 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 the change-of-variables formula of Probability Measures on Euclidean Space and Random Vectors: Standing Notation (iv), applied to the pairing $(\mathrm{id},\mathrm{id})$, Borel by Probability Measures on Euclidean Space and Random Vectors: Standing Notation (v) since $\mathrm{id}$ is Borel (preamble of 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), and to the nonnegative Borel function $f_{q,\eta}$,

$$\int_{\mathbb{R}^{d+d}}f_{q,\eta}\,d\pi=\int_{\mathbb{R}^{d}}f_{q,\eta}\circ(\mathrm{id},\mathrm{id})\,d\nu .$$

For $x\in\mathbb{R}^{d}$, $(\mathrm{id},\mathrm{id})(x)=\iota(x,x)$ by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (ii), and $\mathrm{pr}_{1}(\iota(x,x))=x=\mathrm{pr}_{2}(\iota(x,x))$ by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (i); so $f_{q,\eta}((\mathrm{id},\mathrm{id})(x))=\lVert q(x)-\eta(x)\rVert^{2}$. The pointwise difference $q-\eta$ is a representative of the difference of the two classes, by the operations on classes of The Space of Square-Integrable Random Vectors (ii) (applied on $(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}),\nu)$ as Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation (ii) prescribes) together with $x-y=x+(-1)y$ in $\mathbb{R}^{d}$ (claims 2 and 3 of Euclidean Space $\mathbb{R}^n$ is a Real Vector Space), so $\int_{\mathbb{R}^{d}}\lVert q-\eta\rVert^{2}\,d\nu=\lVert q-\eta\rVert_{\nu}^{2}$ by Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation (ii) and Existence and Uniqueness of the Nonnegative Square Root. This proves claim 2. $\blacksquare$