Proof of The Wasserstein Distance on a Hilbert Space as the Increasing Limit of the Euclidean Wasserstein Distances of the Coordinate Projections

Each result cited below is universally quantified over the data in its own statement and is applied with the data named at the point of citation.

Conventions. For $m\in\mathbb{N}$ and $\alpha,\beta\in\mathcal{P}(\mathbb{R}^{m})$ write $\Pi_{m}(\alpha,\beta)$ for the set of couplings of $\alpha$ and $\beta$ and $I_{m}$ for the quadratic cost of that definition in dimension $m$; by The Quadratic Wasserstein Distance on Euclidean Space §distance, for $\alpha,\beta\in\mathcal{P}_{2}(\mathbb{R}^{m})$ the number $W_{2}^{(m)}(\alpha,\beta)^{2}$ is the greatest lower bound of $\{I_{m}(\gamma):\gamma\in\Pi_{m}(\alpha,\beta)\}$. Likewise $\Pi(\rho,\sigma)$ and $I$ are the couplings and the quadratic cost of that definition, and by The Quadratic Wasserstein Distance on a Hilbert Space §distance the number $W_{2}(\rho,\sigma)^{2}$ is the greatest lower bound of $\{I(\pi):\pi\in\Pi(\rho,\sigma)\}$ for $\rho,\sigma\in\mathcal{P}_{2}(X)$. Hence, by the definition of a greatest lower bound, a real number that is at most the cost of every coupling of two measures is at most their squared distance. Nonnegative square roots preserve the order of nonnegative reals by Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field. The projections $\mathrm{pr}^{q,p}_{1},\mathrm{pr}^{q,p}_{2}$ and the pairing $(u,v)$ of two maps into Euclidean spaces are those of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pairs; the Borel $\sigma$-algebra of $(\mathbb{R}^{q},d_{E})$ is $\mathcal{B}(\mathbb{R}^{q})$ by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §spaces, so the Borel maps and push-forwards of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §measures and Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §pushforward that involve $\mathbb{R}^{q}$ are those of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps and Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward. A composition of Borel maps is Borel by claim 4 of Borel Measurability and Bounded Integration on a Metric Space, and directly from the formula $T_{\#}\rho(B)=\rho(T^{-1}(B))$ of claim 1 of Image Measures, Measures with Densities, and Change of Variables one has $(S\circ T)_{\#}\rho=S_{\#}(T_{\#}\rho)$ whenever $S\circ T$ is defined, because $(S\circ T)^{-1}(B)=T^{-1}(S^{-1}(B))$. Integrals of nonnegative Borel functions are taken in $[0,\infty]$, with the monotonicity and additivity of Linearity and Monotonicity of the Lebesgue Integral §nonnegative, and change of variables under a push-forward is claim 2 of Image Measures, Measures with Densities, and Change of Variables.

By Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §continuity, the maps $p_{n},p_{n}^{*},P_{n},Q_{n}$ are Borel and, for $x,x'\in X$ and $y\in\mathbb{R}^{n}$, $p_{n}(p_{n}^{*}(y))=y$, $|p_{n}^{*}(y)|=\lVert y\rVert$, $|P_{n}x|=\lVert p_{n}(x)\rVert$, $|x|^{2}=|P_{n}x|^{2}+|Q_{n}x|^{2}$ and $\lVert p_{n}(x)-p_{n}(x')\rVert\le|x-x'|$; in particular $|P_{n}x|\le|x|$ and $|Q_{n}x|\le|x|$. Since $P_{n}=p_{n}^{*}\circ p_{n}$ by Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §coordinates, $p_{n}\circ P_{n}=p_{n}\circ p_{n}^{*}\circ p_{n}=p_{n}$. The synthesis map is linear: $p_{n}^{*}(a)-p_{n}^{*}(b)=\sum_{k=1}^{n}(a_{k}-b_{k})e_{k}=p_{n}^{*}(a-b)$, so $|p_{n}^{*}(a)-p_{n}^{*}(b)|=\lVert a-b\rVert$ for $a,b\in\mathbb{R}^{n}$. All of this holds for every $n\in\mathbb{N}$.

Step 1 (Claim 1). Since $p_{n}$ is Borel, $(p_{n})_{\#}\mu$ is a probability measure on $(\mathbb{R}^{n},\mathcal{B}(\mathbb{R}^{n}))$ by claim 1 of Image Measures, Measures with Densities, and Change of Variables, that is, an element of $\mathcal{P}(\mathbb{R}^{n})$. The map $y\mapsto\lVert y\rVert^{2}$ is Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §functions, so by the definition of the second moment, change of variables and monotonicity of the integral,

$$M_{2}((p_{n})_{\#}\mu)=\int_{X}\lVert p_{n}(x)\rVert^{2}\,\mu(dx)=\int_{X}|P_{n}x|^{2}\,\mu(dx)\le\int_{X}|x|^{2}\,\mu(dx)=M_{2}(\mu)<\infty ,$$

the last quantity being the second moment of $\mu\in\mathcal{P}_{2}(X)$. Hence $(p_{n})_{\#}\mu\in\mathcal{P}_{2}(\mathbb{R}^{n})$ by The Second Moment of a Probability Measure on Euclidean Space and the Probability Measures with Finite Second Moment §space. Likewise $P_{n}$ is Borel, so $(P_{n})_{\#}\mu\in\mathcal{P}(X)$, and since $x\mapsto|x|^{2}$ is Borel, as recorded in The Second Moment of a Borel Probability Measure on a Hilbert Space and the Probability Measures with Finite Second Moment, $M_{2}((P_{n})_{\#}\mu)=\int_{X}|P_{n}x|^{2}\,\mu(dx)\le M_{2}(\mu)<\infty$, so $(P_{n})_{\#}\mu\in\mathcal{P}_{2}(X)$ by The Second Moment of a Borel Probability Measure on a Hilbert Space and the Probability Measures with Finite Second Moment §space. The same holds for $\nu$ and for every $n\in\mathbb{N}$, and indeed for every member of $\mathcal{P}_{2}(X)$ in place of $\mu$.

Step 2 (A comparison). We show: for all $\rho,\sigma\in\mathcal{P}_{2}(X)$ and $m\in\mathbb{N}$,

$$W_{2}^{(m)}\bigl((p_{m})_{\#}\rho,(p_{m})_{\#}\sigma\bigr)\le W_{2}(\rho,\sigma),$$

both sides being defined by Step 1. Let $\pi\in\Pi(\rho,\sigma)$. The coordinate maps $\pi_{1},\pi_{2}$ are Borel by Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §product-sigma, so $p_{m}\circ\pi_{1}$ and $p_{m}\circ\pi_{2}$ are measurable with respect to $\mathcal{B}(X\times X)$ and $\mathcal{B}(\mathbb{R}^{m})$, and their pairing $\Psi=(p_{m}\circ\pi_{1},p_{m}\circ\pi_{2}):X\times X\to\mathbb{R}^{m+m}$ is measurable with respect to $\mathcal{B}(X\times X)$ and $\mathcal{B}(\mathbb{R}^{m+m})$ by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §pairing. Thus $\gamma=\Psi_{\#}\pi\in\mathcal{P}(\mathbb{R}^{m+m})$ by claim 1 of Image Measures, Measures with Densities, and Change of Variables. By Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §projections, $\mathrm{pr}^{m,m}_{i}\circ\Psi=p_{m}\circ\pi_{i}$ for $i=1,2$, so $(\mathrm{pr}^{m,m}_{1})_{\#}\gamma=(p_{m})_{\#}((\pi_{1})_{\#}\pi)=(p_{m})_{\#}\rho$ and likewise $(\mathrm{pr}^{m,m}_{2})_{\#}\gamma=(p_{m})_{\#}\sigma$; that is, $\gamma\in\Pi_{m}((p_{m})_{\#}\rho,(p_{m})_{\#}\sigma)$. The function $z\mapsto\lVert\mathrm{pr}^{m,m}_{1}(z)-\mathrm{pr}^{m,m}_{2}(z)\rVert^{2}$ is Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §functions, so change of variables, the inequality $\lVert p_{m}(x)-p_{m}(x')\rVert\le|x-x'|$ of Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §continuity and monotonicity of the integral give

$$I_{m}(\gamma)=\int_{X\times X}\lVert p_{m}(\pi_{1}(z))-p_{m}(\pi_{2}(z))\rVert^{2}\,\pi(dz)\le\int_{X\times X}|\pi_{1}(z)-\pi_{2}(z)|^{2}\,\pi(dz)=I(\pi).$$

Hence $W_{2}^{(m)}((p_{m})_{\#}\rho,(p_{m})_{\#}\sigma)^{2}\le I_{m}(\gamma)\le I(\pi)$ for every $\pi\in\Pi(\rho,\sigma)$, so $W_{2}^{(m)}((p_{m})_{\#}\rho,(p_{m})_{\#}\sigma)^{2}\le W_{2}(\rho,\sigma)^{2}$ by the Conventions, and taking square roots gives the claim of this step.

Step 3 (Claim 2). Put $\alpha=(P_{n})_{\#}\mu$, $\beta=(P_{n})_{\#}\nu$, $\alpha'=(p_{n})_{\#}\mu$ and $\beta'=(p_{n})_{\#}\nu$; by Step 1, $\alpha,\beta\in\mathcal{P}_{2}(X)$ and $\alpha',\beta'\in\mathcal{P}_{2}(\mathbb{R}^{n})$. By the Conventions, $(p_{n})_{\#}\alpha=(p_{n}\circ P_{n})_{\#}\mu=\alpha'$ and $(p_{n}^{*})_{\#}\alpha'=(p_{n}^{*}\circ p_{n})_{\#}\mu=\alpha$, and likewise $(p_{n})_{\#}\beta=\beta'$ and $(p_{n}^{*})_{\#}\beta'=\beta$. Step 2 with $\rho=\alpha$, $\sigma=\beta$ and $m=n$ gives $W_{2}^{(n)}(\alpha',\beta')\le W_{2}(\alpha,\beta)$.

Conversely, let $\gamma\in\Pi_{n}(\alpha',\beta')$. The maps $p_{n}^{*}\circ\mathrm{pr}^{n,n}_{1}$ and $p_{n}^{*}\circ\mathrm{pr}^{n,n}_{2}$ from $\mathbb{R}^{n+n}$ to $X$ are Borel, the projections being Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §projections; hence the map $\Phi=(p_{n}^{*}\circ\mathrm{pr}^{n,n}_{1},p_{n}^{*}\circ\mathrm{pr}^{n,n}_{2}):\mathbb{R}^{n+n}\to X\times X$ of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §pairs is measurable with respect to $\mathcal{B}(\mathbb{R}^{n+n})$ and $\mathcal{B}(X\times X)$ by Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §pairing, and $\pi=\Phi_{\#}\gamma\in\mathcal{P}(X\times X)$ by claim 1 of Image Measures, Measures with Densities, and Change of Variables. Since $\pi_{i}\circ\Phi=p_{n}^{*}\circ\mathrm{pr}^{n,n}_{i}$, we get $(\pi_{1})_{\#}\pi=(p_{n}^{*})_{\#}((\mathrm{pr}^{n,n}_{1})_{\#}\gamma)=(p_{n}^{*})_{\#}\alpha'=\alpha$ and likewise $(\pi_{2})_{\#}\pi=\beta$, so $\pi\in\Pi(\alpha,\beta)$. The integrand of the cost is Borel by Couplings of Two Borel Probability Measures on a Hilbert Space and Their Quadratic Cost §cost, so change of variables and the isometry property of $p_{n}^{*}$ from the Conventions give

$$I(\pi)=\int_{\mathbb{R}^{n+n}}\bigl|p_{n}^{*}(\mathrm{pr}^{n,n}_{1}(w))-p_{n}^{*}(\mathrm{pr}^{n,n}_{2}(w))\bigr|^{2}\,\gamma(dw)=\int_{\mathbb{R}^{n+n}}\lVert\mathrm{pr}^{n,n}_{1}(w)-\mathrm{pr}^{n,n}_{2}(w)\rVert^{2}\,\gamma(dw)=I_{n}(\gamma).$$

Hence $W_{2}(\alpha,\beta)^{2}\le I(\pi)=I_{n}(\gamma)$ for every $\gamma\in\Pi_{n}(\alpha',\beta')$, so $W_{2}(\alpha,\beta)^{2}\le W_{2}^{(n)}(\alpha',\beta')^{2}$ by the Conventions, and $W_{2}(\alpha,\beta)\le W_{2}^{(n)}(\alpha',\beta')$. The two inequalities give claim 2.

Step 4 (Claim 3). The second inequality is Step 2 with $\rho=\mu$, $\sigma=\nu$ and $m=n+1$. For the first, let $D=\mathrm{pr}^{n,1}_{1}:\mathbb{R}^{n+1}\to\mathbb{R}^{n}$, which is Borel with $\lVert D(z)\rVert\le\lVert z\rVert$ by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §projections. By the same claim $z=\iota^{n,1}(D(z),\mathrm{pr}^{n,1}_{2}(z))$, so by the description of $\iota^{n,1}$ in the preamble of Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets the point $D(z)$ is $(z_{1},\dots,z_{n})$. Consequently $D(z)-D(z')=D(z-z')$, so $\lVert D(z)-D(z')\rVert\le\lVert z-z'\rVert$ for $z,z'\in\mathbb{R}^{n+1}$, and $D\circ p_{n+1}=p_{n}$ by the definition of the coordinate maps in Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §coordinates. Put $\alpha''=(p_{n+1})_{\#}\mu$ and $\beta''=(p_{n+1})_{\#}\nu$, members of $\mathcal{P}_{2}(\mathbb{R}^{n+1})$ by Step 1, and let $\gamma\in\Pi_{n+1}(\alpha'',\beta'')$. The pairing $\Delta=(D\circ\mathrm{pr}^{n+1,n+1}_{1},D\circ\mathrm{pr}^{n+1,n+1}_{2}):\mathbb{R}^{(n+1)+(n+1)}\to\mathbb{R}^{n+n}$ is Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §pairing, and $\gamma'=\Delta_{\#}\gamma\in\mathcal{P}(\mathbb{R}^{n+n})$. By Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §projections, $\mathrm{pr}^{n,n}_{i}\circ\Delta=D\circ\mathrm{pr}^{n+1,n+1}_{i}$, so $(\mathrm{pr}^{n,n}_{1})_{\#}\gamma'=D_{\#}\alpha''=(D\circ p_{n+1})_{\#}\mu=(p_{n})_{\#}\mu$ and likewise $(\mathrm{pr}^{n,n}_{2})_{\#}\gamma'=(p_{n})_{\#}\nu$; thus $\gamma'\in\Pi_{n}((p_{n})_{\#}\mu,(p_{n})_{\#}\nu)$. By change of variables (the integrand being Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §functions) and monotonicity,

$$I_{n}(\gamma')=\int\bigl\lVert D(\mathrm{pr}^{n+1,n+1}_{1}(w))-D(\mathrm{pr}^{n+1,n+1}_{2}(w))\bigr\rVert^{2}\,\gamma(dw)\le\int\bigl\lVert\mathrm{pr}^{n+1,n+1}_{1}(w)-\mathrm{pr}^{n+1,n+1}_{2}(w)\bigr\rVert^{2}\,\gamma(dw)=I_{n+1}(\gamma),$$

the integrals being over $\mathbb{R}^{(n+1)+(n+1)}$. Hence $W_{2}^{(n)}((p_{n})_{\#}\mu,(p_{n})_{\#}\nu)^{2}\le I_{n+1}(\gamma)$ for every $\gamma\in\Pi_{n+1}(\alpha'',\beta'')$, so $W_{2}^{(n)}((p_{n})_{\#}\mu,(p_{n})_{\#}\nu)^{2}\le W_{2}^{(n+1)}(\alpha'',\beta'')^{2}$ by the Conventions, and taking square roots gives the first inequality.

Step 5 (Claim 4). By Step 1, $(P_{n})_{\#}\mu\in\mathcal{P}_{2}(X)$. The maps $P_{n}$ and $\mathrm{id}_{X}$ are Borel, so by Couplings on a Hilbert Space: Product Coupling, Swap, Finiteness of the Cost, Push-Forward Couplings, Modifying One Marginal, Quantisation, Gluing over a Finitely Supported Measure, the Lipschitz Bound and the Moment Bound §pushforward with $S=P_{n}$ and $T=\mathrm{id}_{X}$ the measure $\pi=(P_{n},\mathrm{id}_{X})_{\#}\mu$ belongs to $\Pi((P_{n})_{\#}\mu,\mu)$ and $I(\pi)=\int_{X}|P_{n}x-x|^{2}\,\mu(dx)=\int_{X}|Q_{n}x|^{2}\,\mu(dx)$, since $P_{n}x-x=-Q_{n}x$. Hence $W_{2}((P_{n})_{\#}\mu,\mu)^{2}\le I(\pi)=\int_{X}|Q_{n}x|^{2}\,\mu(dx)$ by The Quadratic Wasserstein Distance on a Hilbert Space §distance.

For the limit, put $f_{m}(x)=|Q_{m}x|^{2}=|x-P_{m}x|^{2}$ for $m\in\mathbb{N}$ and $x\in X$; each $f_{m}:X\to\mathbb{R}$ is Borel by the preamble of Couplings on a Hilbert Space: Product Coupling, Swap, Finiteness of the Cost, Push-Forward Couplings, Modifying One Marginal, Quantisation, Gluing over a Finitely Supported Measure, the Lipschitz Bound and the Moment Bound, applied with $S=\mathrm{id}_{X}$ and $T=P_{m}$. By Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §coordinates, $(X_{m})_{m\in\mathbb{N}}$ is an exhausting sequence for $X$, $P_{m}$ is the orthogonal projection onto $X_{m}$ and $Q_{m}x=x-P_{m}x$; so Exhausting Sequences of Finite-Dimensional Subspaces in a Separable Real Hilbert Space, and Their Projections §tail, applied with $H=X$ and $H_{m}=X_{m}$, shows that $(Q_{m}x)_{m}$ converges to $0_{X}$ in $(X,d)$ for every $x\in X$. By The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §continuity the real sequence $(|Q_{m}x|)_{m}$ converges to $|0_{X}|=0$, and by claim 2 of Arithmetic of Limits of Real Sequences the sequence $(f_{m}(x))_{m}$ converges to $0$. Moreover $|f_{m}(x)|\le|x|^{2}$ for all $m$ and $x$, and $g(x)=|x|^{2}$ is a Borel function with $\int_{X}|g|\,d\mu=M_{2}(\mu)<\infty$, hence integrable with respect to $\mu$ by the criterion recorded in that definition. Since $0\le f_{m}\le g$, monotonicity of the integral of nonnegative functions gives $\int_{X}|f_{m}|\,d\mu\le\int_{X}g\,d\mu<\infty$, so each $f_{m}$ is integrable by the same criterion and $\int_{X}f_{m}\,d\mu$ is a real number. Claim 3 of Dominated Convergence Theorem, applied on the measure space $(X,\mathcal{B}(X),\mu)$ to $(f_{m})_{m}$, the limit function $0$ and the dominating function $g$, shows that $(\int_{X}f_{m}\,d\mu)_{m}$ converges to $\int_{X}0\,d\mu=0$. As each $f_{m}$ is nonnegative, its negative part vanishes, so by Integrable Function and the Lebesgue Integral its integral as an integrable function is its integral $\int_{X}|Q_{m}x|^{2}\,\mu(dx)$ in $[0,\infty]$. This proves claim 4 (for every member of $\mathcal{P}_{2}(X)$ in place of $\mu$, in particular for $\nu$).

Step 6 (Claim 5). Put $a_{m}=W_{2}^{(m)}((p_{m})_{\#}\mu,(p_{m})_{\#}\nu)$ and $w=W_{2}(\mu,\nu)$. By Step 4 applied at each index $m$, $a_{m}\le a_{m+1}\le w$ for every $m\in\mathbb{N}$, so the set $A=\{a_{m}:m\in\mathbb{N}\}$ is bounded above by $w$, and by A Bounded Monotone Sequence of Real Numbers Converges §nondecreasing its supremum $s=\sup A$ exists and $(a_{m})_{m}$ converges to $s$; moreover $s\le w$, since $w$ is an upper bound of $A$.

It remains to show $w\le s$. Let $\varepsilon>0$. By Step 5 applied to $\mu$ and to $\nu$ and by the definition of the limit, there is $n\in\mathbb{N}$ (the larger of the two indices provided) with $\int_{X}|Q_{n}x|^{2}\,\mu(dx)<\varepsilon^{2}/4$ and $\int_{X}|Q_{n}x|^{2}\,\nu(dx)<\varepsilon^{2}/4$. By Step 5, $W_{2}((P_{n})_{\#}\mu,\mu)^{2}<\varepsilon^{2}/4$ and $W_{2}((P_{n})_{\#}\nu,\nu)^{2}<\varepsilon^{2}/4$, so both distances are less than $\varepsilon/2$. All of $\mu,\nu,(P_{n})_{\#}\mu,(P_{n})_{\#}\nu$ lie in $\mathcal{P}_{2}(X)$ by Step 1, so The Quadratic Wasserstein Distance is a Metric on the Probability Measures with Finite Second Moment on a Hilbert Space §triangle, applied twice, and The Quadratic Wasserstein Distance is a Metric on the Probability Measures with Finite Second Moment on a Hilbert Space §symmetry give

$$w\le W_{2}(\mu,(P_{n})_{\#}\mu)+W_{2}((P_{n})_{\#}\mu,(P_{n})_{\#}\nu)+W_{2}((P_{n})_{\#}\nu,\nu)<\tfrac{\varepsilon}{2}+a_{n}+\tfrac{\varepsilon}{2}\le s+\varepsilon ,$$

where the middle term equals $a_{n}$ by Step 3 and $a_{n}\le s$. As $\varepsilon>0$ was arbitrary, $w\le s$. Hence $s=w$, and $(a_{m})_{m}$ converges to $W_{2}(\mu,\nu)$. $\blacksquare$