Proof of Measures Whose Heads Agree with a Fixed Measure and Whose Tails Vanish Converge in the Quadratic Wasserstein Distance

Each result cited is universally quantified over the data in its own statement.

Elementary order and arithmetic of real numbers is carried by The Real Numbers: Standing Notation and Background §background, in force through Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §background. For $n\in\mathbb{N}$ write

$$t_{n}=\int_{X}|Q_{n}x|^{2}\,\lambda_{n}(dx),\qquad s_{n}=\int_{X}|Q_{n}y|^{2}\,\nu(dy),$$

nonnegative real numbers by the integrability recorded in the statement, since $\lambda_{n},\nu\in\mathcal{P}_{2}(X)$.

Step 1 (the head projections). Let $n\in\mathbb{N}$. By Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §continuity the maps $p_{n}$, $p_{n}^{*}$ and $P_{n}=p_{n}^{*}\circ p_{n}$ (Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §coordinates) are Lipschitz, hence Borel. For every $\lambda\in\mathcal{P}(X)$ and every Borel set $B\subseteq X$ one has $P_{n}^{-1}(B)=p_{n}^{-1}\bigl((p_{n}^{*})^{-1}(B)\bigr)$, so, with the push-forwards of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §pushforward,

$$(P_{n})_{\#}\lambda(B)=\bigl((p_{n})_{\#}\lambda\bigr)\bigl((p_{n}^{*})^{-1}(B)\bigr)=\bigl((p_{n}^{*})_{\#}((p_{n})_{\#}\lambda)\bigr)(B).$$

Thus $(P_{n})_{\#}\lambda$ depends only on $(p_{n})_{\#}\lambda$, and the hypothesis $(p_{n})_{\#}\lambda_{n}=(p_{n})_{\#}\nu$ gives

$$\nu_{n}:=(P_{n})_{\#}\lambda_{n}=(P_{n})_{\#}\nu .$$

Step 2 (distance to the head projection). Let $\lambda\in\mathcal{P}_{2}(X)$ and $n\in\mathbb{N}$. The identity map $\mathrm{id}_{X}$ is continuous, hence Borel, and $P_{n}$ is Borel by Step 1, 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, applied with $S=\mathrm{id}_{X}$ and $T=P_{n}$, the coupling $(\mathrm{id}_{X},P_{n})_{\#}\lambda$ belongs to $\Pi(\lambda,(P_{n})_{\#}\lambda)$, since $(\mathrm{id}_{X})_{\#}\lambda=\lambda$, and has quadratic cost

$$\int_{X}|x-P_{n}x|^{2}\,\lambda(dx)=\int_{X}|Q_{n}x|^{2}\,\lambda(dx),$$

as $Q_{n}x=x-P_{n}x$ by Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §coordinates. This cost is finite, so $(P_{n})_{\#}\lambda\in\mathcal{P}_{2}(X)$ by the converse part 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 §cost-finite, and The Quadratic Wasserstein Distance on a Hilbert Space §distance gives

$$W_{2}\bigl(\lambda,(P_{n})_{\#}\lambda\bigr)^{2}\le\int_{X}|Q_{n}x|^{2}\,\lambda(dx).$$

Applied with $\lambda=\lambda_{n}$ and with $\lambda=\nu$, and using Step 1 and the symmetry The Quadratic Wasserstein Distance is a Metric on the Probability Measures with Finite Second Moment on a Hilbert Space §symmetry, this shows $\nu_{n}\in\mathcal{P}_{2}(X)$,

$$W_{2}(\lambda_{n},\nu_{n})^{2}\le t_{n},\qquad W_{2}(\nu_{n},\nu)^{2}=W_{2}(\nu,\nu_{n})^{2}\le s_{n}. \tag{1}$$

Step 3 (the tails of $\nu$ vanish). Let $y\in X$. The sequence $(X_{n})_{n\in\mathbb{N}}$ is exhausting with projections $P_{n}$ and $Q_{n}$ (Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §coordinates), so by Exhausting Sequences of Finite-Dimensional Subspaces in a Separable Real Hilbert Space, and Their Projections §tail the sequence $(Q_{n}y)_{n\in\mathbb{N}}$ converges to $0_{X}$ in $(X,d)$; that is, $|Q_{n}y|=d(Q_{n}y,0_{X})$ converges to $0$, and hence so does $|Q_{n}y|^{2}$. Each function $y\mapsto|Q_{n}y|^{2}$ is Borel and satisfies $0\le|Q_{n}y|^{2}\le|y|^{2}$, by the statement and Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §continuity, and $y\mapsto|y|^{2}$ is integrable with respect to $\nu$, because $\nu\in\mathcal{P}_{2}(X)$ (The Second Moment of a Borel Probability Measure on a Hilbert Space and the Probability Measures with Finite Second Moment §space). Claim 3 of Dominated Convergence Theorem, applied on $(X,\mathcal{B}(X),\nu)$ with $f_{n}(y)=|Q_{n}y|^{2}$, limit $f=0$ and dominating function $g(y)=|y|^{2}$, gives $\lim_{n\to\infty}s_{n}=0$.

Step 4 (conclusion). Let $\varepsilon\in\mathbb{R}$ be positive. By the hypothesis $\lim_{n}t_{n}=0$ and by Step 3, both with the positive number $\varepsilon^{2}/4$, there are $N_{1},N_{2}\in\mathbb{N}$ with $t_{n}<\varepsilon^{2}/4$ for $n\ge N_{1}$ and $s_{n}<\varepsilon^{2}/4$ for $n\ge N_{2}$; let $N$ be the larger of $N_{1}$ and $N_{2}$, and let $n\ge N$. By (1), $W_{2}(\lambda_{n},\nu_{n})^{2}<(\varepsilon/2)^{2}$ and $W_{2}(\nu_{n},\nu)^{2}<(\varepsilon/2)^{2}$; all four numbers being nonnegative, claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field gives $W_{2}(\lambda_{n},\nu_{n})<\varepsilon/2$ and $W_{2}(\nu_{n},\nu)<\varepsilon/2$. Since $\lambda_{n},\nu_{n},\nu\in\mathcal{P}_{2}(X)$, the triangle inequality The Quadratic Wasserstein Distance is a Metric on the Probability Measures with Finite Second Moment on a Hilbert Space §triangle gives

$$0\le W_{2}(\lambda_{n},\nu)\le W_{2}(\lambda_{n},\nu_{n})+W_{2}(\nu_{n},\nu)<\varepsilon .$$

As $\varepsilon$ was arbitrary, $\lim_{n\to\infty}W_{2}(\lambda_{n},\nu)=0$ in the sense of Limit of a Sequence of Real Numbers. $\blacksquare$