Proof of The Optimal Map as a Square-Integrable Vector Field: Integrability, Transport Cost and Uniqueness of the Class

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.

Claim 1. Let $S:\mathbb{R}^{d}\to\mathbb{R}^{d}$ be Borel. The function $\lVert S\rVert^{2}$ 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 $(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}))$ with $S$ in the role of both $u$ and $v$ there, and it is nonnegative by Nonnegativity of Squares in an Ordered Field, being a square. Likewise the function $\mathbb{R}^{d}\to\mathbb{R}$ with value $\lVert x\rVert^{2}$ at $x$ is Borel and nonnegative, as recorded in Probability Measures on Euclidean Space and Random Vectors: Standing Notation (v). Both are read as maps into $[0,\infty]$, the two readings of measurability agreeing by Probability Measures on Euclidean Space and Random Vectors: Standing Notation (iii).

By The Second Moment of a Probability Measure on Euclidean Space and the Probability Measures with Finite Second Moment (i), read with $S_{\#}\mu$ in place of the measure named there,

$$M_{2}(S_{\#}\mu)=\int_{\mathbb{R}^{d}}\lVert x\rVert^{2}\,(S_{\#}\mu)(dx).$$

The change-of-variables formula of Probability Measures on Euclidean Space and Random Vectors: Standing Notation (iv), applied to the Borel map $S$ and to the nonnegative Borel function $x\mapsto\lVert x\rVert^{2}$, turns the right-hand side into $\int_{\mathbb{R}^{d}}\lVert S\rVert^{2}\,d\mu$, the composite of $x\mapsto\lVert x\rVert^{2}$ with $S$ being the function $\lVert S\rVert^{2}$. This proves the displayed identity, both sides being elements of $[0,\infty]$.

Suppose now that $\int_{\mathbb{R}^{d}}\lVert S\rVert^{2}\,d\mu<\infty$. Then $M_{2}(S_{\#}\mu)<\infty$ by that identity, and $S_{\#}\mu\in\mathcal{P}(\mathbb{R}^{d})$ by Probability Measures on Euclidean Space and Random Vectors: Standing Notation (iv), so $S_{\#}\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ by The Second Moment of a Probability Measure on Euclidean Space and the Probability Measures with Finite Second Moment (ii). Being a Borel map $\mathbb{R}^{d}\to\mathbb{R}^{d}$ whose squared norm has finite integral against $\mu$, $S$ has a class in $L^{2}(\mu;\mathbb{R}^{d})$ by Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation (ii). That clause also records that $L^{2}(\mu;\mathbb{R}^{d})$ is a real Hilbert space, hence a real vector space, and $\mathrm{id}$ belongs to it 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); therefore the difference $\mathrm{id}-S$ belongs to it.

Suppose instead that $S_{\#}\mu=\nu$. Then the displayed identity gives $\int_{\mathbb{R}^{d}}\lVert S\rVert^{2}\,d\mu=M_{2}(\nu)$, which is finite because $\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$, by The Second Moment of a Probability Measure on Euclidean Space and the Probability Measures with Finite Second Moment (ii); so the hypothesis of the previous paragraph holds and its conclusions follow.

Finally let $a\in\mathbb{R}^{d}$, so that $\tau_{a}$ is the Borel map $x\mapsto x+a$ of The Wasserstein Space and Its Lift to Square-Integrable Random Vectors: Standing Notation (vi). Fix $x\in\mathbb{R}^{d}$. By the triangle inequality, claim 6 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square, $\lVert x+a\rVert\le\lVert x\rVert+\lVert a\rVert$; both sides are nonnegative by claim 1 of that lemma and by claim 2 of Elementary Arithmetic in an Ordered Field, so squaring preserves the inequality, by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, and

$$\lVert x+a\rVert^{2}\le\bigl(\lVert x\rVert+\lVert a\rVert\bigr)^{2}=\lVert x\rVert^{2}+2\,\lVert x\rVert\,\lVert a\rVert+\lVert a\rVert^{2},$$

the equality being claim 5 of Zero Products and Elementary Identities in a Field. By the inequality $\lVert x\rVert\,\lVert a\rVert\le\tfrac{1}{2}(\lVert x\rVert^{2}+\lVert a\rVert^{2})$ recorded in Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (iv), multiplied by the nonnegative $2$ (claim 5 of Elementary Arithmetic in an Ordered Field, the positivity of $2$ being claim 8 of Elementary Order Arithmetic in an Ordered Field) and added to the preceding bound by claim 3 of Elementary Arithmetic in an Ordered Field,

$$\lVert\tau_{a}(x)\rVert^{2}\le2\,\lVert x\rVert^{2}+2\,\lVert a\rVert^{2}\qquad\text{for every }x\in\mathbb{R}^{d}.$$

The function on the right is nonnegative and Borel, being the sum of a real scalar multiple of the Borel function $x\mapsto\lVert x\rVert^{2}$ and the constant function with value $2\lVert a\rVert^{2}$, which is measurable by claim 1 of Arithmetic of Measurable Real- and [0,∞]-Valued Maps, the sum and the scalar multiple being measurable by claim 2 of that lemma. Integrating the pointwise bound against $\mu$ and using claim 1 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion for monotonicity, additivity and nonnegative scalar multiples,

$$\int_{\mathbb{R}^{d}}\lVert\tau_{a}\rVert^{2}\,d\mu\le2\int_{\mathbb{R}^{d}}\lVert x\rVert^{2}\,\mu(dx)+2\,\lVert a\rVert^{2}\int_{\mathbb{R}^{d}}\mathbf{1}_{\mathbb{R}^{d}}\,d\mu=2\,M_{2}(\mu)+2\,\lVert a\rVert^{2},$$

the constant function with value $2\lVert a\rVert^{2}$ being $2\lVert a\rVert^{2}\,\mathbf{1}_{\mathbb{R}^{d}}$, and $\int_{\mathbb{R}^{d}}\mathbf{1}_{\mathbb{R}^{d}}\,d\mu=\mu(\mathbb{R}^{d})=1$ by The Integral of an Indicator Function is the Measure of the Set and Probability Measures on Euclidean Space and Random Vectors: Standing Notation (iii). The right-hand side is a real number, since $M_{2}(\mu)<\infty$; so the hypothesis of the second paragraph holds for $\tau_{a}$, and $(\tau_{a})_{\#}\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ with the stated bound, the bound being the displayed inequality read through the identity of the first paragraph.

Claim 2. Let $T$ be an optimal map from $\mu$ to $\nu$. By Optimal Transport Maps and Uniquely Mapped Pairs of Probability Measures (i) one has $T_{\#}\mu=\nu$ and the coupling $(\mathrm{id},T)_{\#}\mu$ is optimal, so $I((\mathrm{id},T)_{\#}\mu)=W_{2}(\mu,\nu)^{2}$ by Optimal Coupling of Two Probability Measures with Finite Second Moment (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 $T$ in the roles of the two Borel maps there,

$$I\bigl((\mathrm{id},T)_{\#}\mu\bigr)=\int_{\mathbb{R}^{d}}\lVert x-T(x)\rVert^{2}\,\mu(dx).$$

By claim 1 the class of $\mathrm{id}-T$ belongs to $L^{2}(\mu;\mathbb{R}^{d})$, and the map $x\mapsto x-T(x)$ represents it, so its norm satisfies $\lVert\mathrm{id}-T\rVert_{\mu}^{2}=\int_{\mathbb{R}^{d}}\lVert x-T(x)\rVert^{2}\,\mu(dx)$ by Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation (ii). Combining the three displays gives $\lVert\mathrm{id}-T\rVert_{\mu}^{2}=W_{2}(\mu,\nu)^{2}$.

Claim 3. Since $(\mu,\nu)$ is uniquely mapped, Optimal Transport Maps and Uniquely Mapped Pairs of Probability Measures (ii) supplies an optimal map $T_{0}$ from $\mu$ to $\nu$ such that every optimal coupling of $\mu$ and $\nu$ equals $(\mathrm{id},T_{0})_{\#}\mu$. The maps $T$ and $T'$ are optimal, so $(\mathrm{id},T)_{\#}\mu$ and $(\mathrm{id},T')_{\#}\mu$ are optimal couplings of $\mu$ and $\nu$ by Optimal Transport Maps and Uniquely Mapped Pairs of Probability Measures (i); both therefore equal $(\mathrm{id},T_{0})_{\#}\mu$, and in particular they are equal to each other. Write $\pi$ for this common measure.

The maps $T\circ\mathrm{pr}_{1}$ and $\mathrm{pr}_{2}$ from $\mathbb{R}^{d+d}$ to $\mathbb{R}^{d}$ are Borel, the projections by Probability Measures on Euclidean Space and Random Vectors: Standing Notation (v) and the composite by Probability Measures on Euclidean Space and Random Vectors: Standing Notation (ii), so the function

$$g:\mathbb{R}^{d+d}\to\mathbb{R},\qquad g(z)=\bigl\lVert T(\mathrm{pr}_{1}(z))-\mathrm{pr}_{2}(z)\bigr\rVert^{2},$$

is Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (iv) and nonnegative by Nonnegativity of Squares in an Ordered Field. We compute $\int_{\mathbb{R}^{d+d}}g\,d\pi$ through each of the two representations of $\pi$, using the change-of-variables formula of Probability Measures on Euclidean Space and Random Vectors: Standing Notation (iv) and the identities $\mathrm{pr}_{1}((\mathrm{id},R)(x))=x$ and $\mathrm{pr}_{2}((\mathrm{id},R)(x))=R(x)$ of Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (i), valid for any Borel $R:\mathbb{R}^{d}\to\mathbb{R}^{d}$. With $\pi=(\mathrm{id},T)_{\#}\mu$ the integrand becomes $\lVert T(x)-T(x)\rVert^{2}$, which is $0$ for every $x$ by claim 3 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square, so the integral is $0$ by claim 1 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion applied with the constant $0$. With $\pi=(\mathrm{id},T')_{\#}\mu$ it becomes $\lVert T(x)-T'(x)\rVert^{2}$. Hence

$$\int_{\mathbb{R}^{d}}\lVert T(x)-T'(x)\rVert^{2}\,\mu(dx)=0 .$$

By claim 1 the classes of $T$ and of $T'$ belong to $L^{2}(\mu;\mathbb{R}^{d})$, and $x\mapsto T(x)-T'(x)$ represents their difference, so the integral just displayed is $\lVert T-T'\rVert_{\mu}^{2}$ by Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation (ii). Therefore $\lVert T-T'\rVert_{\mu}=0$, since a nonnegative real number whose square is $0$ is $0$ by Existence and Uniqueness of the Nonnegative Square Root, and $T-T'$ is the zero class by Elementary Identities in a Real Inner Product Space (iii); that is, the classes of $T$ and $T'$ are equal. Finally $(\mathrm{id}-T)-(\mathrm{id}-T')=T'-T$ in the real vector space $L^{2}(\mu;\mathbb{R}^{d})$, which is the zero class, so the classes of $\mathrm{id}-T$ and of $\mathrm{id}-T'$ are equal as well.