Proof of The First Marginal of a Probability Measure on a Product: Finite Second Moment, the Wasserstein Contraction, and the Gradient Plan

Each result cited is universally quantified over the data in its own statement. Write $\mathrm{pr}_{1}^{d,k}$ for the coordinate projection $\mathbb{R}^{d+k}\to\mathbb{R}^{d}$ and $\mathrm{pr}_{1},\mathrm{pr}_{2}$ for the coordinate projections of $\mathbb{R}^{d+d}$, all Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (i), which also gives $\lVert\mathrm{pr}_{1}^{d,k}(w)\rVert\le\lVert w\rVert$ for $w\in\mathbb{R}^{d+k}$ and $u=\iota(\mathrm{pr}_{1}^{d,k}(u),\mathrm{pr}_{2}^{d,k}(u))$ for $u\in\mathbb{R}^{d+k}$. Since both sides of the first inequality are nonnegative, claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field turns it into $\lVert\mathrm{pr}_{1}^{d,k}(w)\rVert^{2}\le\lVert w\rVert^{2}$, and we use this form below without further comment; the same applies to the projections of $\mathbb{R}^{d+d}$.

Claim 1. By Probability Measures on Euclidean Space and Random Vectors: Standing Notation (iv) the push-forward $\gamma^{(1)}$ belongs to $\mathcal{P}(\mathbb{R}^{d})$, and the change-of-variables formula there, applied to the nonnegative Borel function $y\mapsto\lVert y\rVert^{2}$ of Probability Measures on Euclidean Space and Random Vectors: Standing Notation (v), together with the monotonicity of the integral of nonnegative functions in claim 1 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion, gives

$$M_{2}\bigl(\gamma^{(1)}\bigr)=\int_{\mathbb{R}^{d+k}}\bigl\lVert\mathrm{pr}_{1}^{d,k}(w)\bigr\rVert^{2}\,\gamma(dw)\le\int_{\mathbb{R}^{d+k}}\lVert w\rVert^{2}\,\gamma(dw)=M_{2}(\gamma)<\infty ,$$

the last inequality because $\gamma\in\mathcal{P}_{2}(\mathbb{R}^{d+k})$. Hence $\gamma^{(1)}\in\mathcal{P}_{2}(\mathbb{R}^{d})$ by The Second Moment of a Probability Measure on Euclidean Space and the Probability Measures with Finite Second Moment (ii), proving claim 1.

Claim 2. First, the projection $\mathrm{pr}_{1}^{d,k}$ respects differences: for $u,v\in\mathbb{R}^{d+k}$ and $i\in[d]$, the $i$th component of $\mathrm{pr}_{1}^{d,k}(u)$ is $u_{i}$, by the coordinate description of the concatenation map recorded in the preamble of Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets together with the identity $u=\iota(\mathrm{pr}_{1}^{d,k}(u),\mathrm{pr}_{2}^{d,k}(u))$; since differences in a Euclidean space are taken componentwise, by claim 1 of Difference, Dot Product, and Orthogonality in $\mathbb{R}^n$, the $i$th components of $\mathrm{pr}_{1}^{d,k}(u-v)$ and of $\mathrm{pr}_{1}^{d,k}(u)-\mathrm{pr}_{1}^{d,k}(v)$ are both $u_{i}-v_{i}$, so the two points are equal by claim 1 of Euclidean Points as Tuples of Real Numbers.

Let $\Gamma\in\Pi(\gamma,\gamma')$, a nonempty set 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 (i), the couplings being those of $\mathcal{P}(\mathbb{R}^{d+k})$ and hence measures on $\mathbb{R}^{(d+k)+(d+k)}$; write $\mathrm{PR}_{1},\mathrm{PR}_{2}:\mathbb{R}^{(d+k)+(d+k)}\to\mathbb{R}^{d+k}$ for its coordinate projections, so that $(\mathrm{PR}_{1})_{\#}\Gamma=\gamma$ and $(\mathrm{PR}_{2})_{\#}\Gamma=\gamma'$ by Couplings of Two Probability Measures on Euclidean Space and Their Quadratic Cost (i). Let

$$S=\bigl(\mathrm{pr}_{1}^{d,k}\circ\mathrm{PR}_{1},\ \mathrm{pr}_{1}^{d,k}\circ\mathrm{PR}_{2}\bigr):\mathbb{R}^{(d+k)+(d+k)}\to\mathbb{R}^{d+d},$$

a Borel map, being the pairing of two compositions of Borel maps, which are Borel by Probability Measures on Euclidean Space and Random Vectors: Standing Notation (ii).

By Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (i) together with the description 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) one has $\mathrm{pr}_{1}\circ S=\mathrm{pr}_{1}^{d,k}\circ\mathrm{PR}_{1}$ and $\mathrm{pr}_{2}\circ S=\mathrm{pr}_{1}^{d,k}\circ\mathrm{PR}_{2}$, so for $B\in\mathcal{B}(\mathbb{R}^{d})$, using the description of the push-forward in Probability Measures on Euclidean Space and Random Vectors: Standing Notation (iv) and the identity $(A\circ C)^{-1}(B)=C^{-1}(A^{-1}(B))$ for preimages,

$$\bigl((\mathrm{pr}_{1})_{\#}S_{\#}\Gamma\bigr)(B)=\Gamma\bigl((\mathrm{pr}_{1}^{d,k}\circ\mathrm{PR}_{1})^{-1}(B)\bigr)=\bigl((\mathrm{pr}_{1}^{d,k})_{\#}\gamma\bigr)(B)=\gamma^{(1)}(B),$$

and likewise $(\mathrm{pr}_{2})_{\#}S_{\#}\Gamma=\gamma'^{(1)}$. Hence $S_{\#}\Gamma\in\Pi(\gamma^{(1)},\gamma'^{(1)})$ by Couplings of Two Probability Measures on Euclidean Space and Their Quadratic Cost (i).

By Couplings of Two Probability Measures on Euclidean Space and Their Quadratic Cost (ii), the change-of-variables formula, the identity of the first paragraph of this claim and the monotonicity of the integral of nonnegative functions,

$$I\bigl(S_{\#}\Gamma\bigr)=\int\bigl\lVert\mathrm{pr}_{1}^{d,k}\bigl(\mathrm{PR}_{1}(w)-\mathrm{PR}_{2}(w)\bigr)\bigr\rVert^{2}\,\Gamma(dw)\le\int\bigl\lVert\mathrm{PR}_{1}(w)-\mathrm{PR}_{2}(w)\bigr\rVert^{2}\,\Gamma(dw)=I(\Gamma).$$

By The Quadratic Wasserstein Distance on Euclidean Space (i) one has $W_{2}(\gamma^{(1)},\gamma'^{(1)})^{2}\le I(S_{\#}\Gamma)$, hence $W_{2}(\gamma^{(1)},\gamma'^{(1)})^{2}\le I(\Gamma)$. As $\Gamma$ was an arbitrary member of $\Pi(\gamma,\gamma')$, the real number $W_{2}(\gamma^{(1)},\gamma'^{(1)})^{2}$ is a lower bound of $\{I(\Gamma):\Gamma\in\Pi(\gamma,\gamma')\}$, hence is at most its greatest lower bound, which is a lower bound of that set no smaller than any other by the definition of a greatest lower bound, and which is $W_{2}(\gamma,\gamma')^{2}$ by The Quadratic Wasserstein Distance on Euclidean Space (i) and Existence and Uniqueness of the Nonnegative Square Root. Both distances being nonnegative, claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field gives claim 2.

Claim 3. Fix a representative of $\eta$, a Borel map $\mathbb{R}^{d+k}\to\mathbb{R}^{d+k}$ with $\int\lVert\eta\rVert^{2}\,d\gamma<\infty$, as in Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation (ii). The composition $\mathrm{pr}_{1}^{d,k}\circ\eta$ is Borel by Probability Measures on Euclidean Space and Random Vectors: Standing Notation (ii), so $\Xi^{\eta}$ is Borel by Probability Measures on Euclidean Space and Random Vectors: Standing Notation (v) and $\Xi^{\eta}_{\#}\gamma\in\mathcal{P}(\mathbb{R}^{d+d})$ by Probability Measures on Euclidean Space and Random Vectors: Standing Notation (iv).

By claim 3 of Concatenation Identifies a Product of Euclidean Spaces with a Euclidean Space and the bound on the projection,

$$\bigl\lVert\Xi^{\eta}(w)\bigr\rVert^{2}=\bigl\lVert\mathrm{pr}_{1}^{d,k}(w)\bigr\rVert^{2}+\bigl\lVert\mathrm{pr}_{1}^{d,k}(\eta(w))\bigr\rVert^{2}\le\lVert w\rVert^{2}+\lVert\eta(w)\rVert^{2},$$

so the change-of-variables formula, the additivity and monotonicity of the integral of nonnegative functions in claim 1 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion, and claim 1 give

$$M_{2}\bigl(\Xi^{\eta}_{\#}\gamma\bigr)=\int\bigl\lVert\Xi^{\eta}(w)\bigr\rVert^{2}\,\gamma(dw)\le M_{2}(\gamma)+\lVert\eta\rVert_{\gamma}^{2}<\infty ,$$

so $\Xi^{\eta}_{\#}\gamma$ is a plan in the sense of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation (iv). Exactly as in claim 2, $\mathrm{pr}_{1}\circ\Xi^{\eta}=\mathrm{pr}_{1}^{d,k}$ by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (i) and Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (ii), so its first marginal is $(\mathrm{pr}_{1}^{d,k})_{\#}\gamma=\gamma^{(1)}$. Applying the change-of-variables formula to the nonnegative Borel function $z\mapsto\lVert p\rVert^{2}$, whose composition with $\Xi^{\eta}$ is $w\mapsto\lVert\mathrm{pr}_{1}^{d,k}(\eta(w))\rVert^{2}$ by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (i), and using the bound on the projection once more,

$$\int\lVert p\rVert^{2}\,d\bigl(\Xi^{\eta}_{\#}\gamma\bigr)\le\int\lVert\eta(w)\rVert^{2}\,\gamma(dw)=\lVert\eta\rVert_{\gamma}^{2}.$$

Finally, let $\tilde{\eta}$ be a second representative of the same class, so that the set $S$ on which $\eta$ and $\tilde{\eta}$ agree satisfies $\gamma(S)=1$; it belongs to $\mathcal{B}(\mathbb{R}^{d+k})$, being the preimage of the Borel set $\{0_{\mathbb{R}^{d+k}}\}$ of Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (v) under the difference $\eta-\tilde{\eta}$, which is Borel because each of its components is a difference of Borel real-valued functions by claim 2 of Arithmetic of Measurable Real- and [0,∞]-Valued Maps and claim 2 of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets. Its complement $N$ satisfies $\gamma(N)=\gamma(\mathbb{R}^{d+k})-\gamma(S)=0$ by claim 3 of Basic Properties of a Measure, and the maps $\Xi^{\eta}$ and $\Xi^{\tilde{\eta}}$ agree on $S$.

Let $B\in\mathcal{B}(\mathbb{R}^{d+d})$ and put $A=(\Xi^{\eta})^{-1}(B)$. Since $A\cap N\subseteq N$, the monotonicity of a measure in claim 2 of Basic Properties of a Measure gives $\gamma(A\cap N)=0$, the values of a measure being nonnegative; since $A\cap N\subseteq A$ and $\gamma$ is finite, claim 3 of Basic Properties of a Measure gives $\gamma(A\setminus N)=\gamma(A)-\gamma(A\cap N)=\gamma(A)$, where $A\setminus N=A\setminus(A\cap N)$. On $A\setminus N$ the two maps agree, so $A\setminus N\subseteq(\Xi^{\tilde{\eta}})^{-1}(B)$, and monotonicity gives $\gamma(A)\le\gamma((\Xi^{\tilde{\eta}})^{-1}(B))$; by symmetry the two are equal, so the two push-forwards agree. This proves claim 3.