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 §projections, 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 §pushforward 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 §pairs, together with the monotonicity of the integral of nonnegative functions in claim 1 of Linearity and Monotonicity of the Lebesgue Integral, gives

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 §space, 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 §product, 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 §coupling. Let

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 §borel-maps.

By Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §projections 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 §pairing 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 §pushforward and the identity $(A\circ C)^{-1}(B)=C^{-1}(A^{-1}(B))$ for preimages,

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 §coupling.

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

By The Quadratic Wasserstein Distance on Euclidean Space §distance 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 §distance 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 Plans, Marginals, Vector Fields and Symmetric Matrices on the Wasserstein Space: Standing Notation §fields. The composition $\mathrm{pr}_{1}^{d,k}\circ\eta$ is Borel by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps, so $\Xi^{\eta}$ is Borel by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pairs and $\Xi^{\eta}_{\#}\gamma\in\mathcal{P}(\mathbb{R}^{d+d})$ by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward.

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

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 claim 1 give

so $\Xi^{\eta}_{\#}\gamma$ is a plan in the sense of Plans, Marginals, Vector Fields and Symmetric Matrices on the Wasserstein Space: Standing Notation §plans. 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 §projections and Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §pairing, 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 §projections, and using the bound on the projection once more,

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 §finite-sets 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, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions 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.