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

In the setting of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation, with the maps $p_{n},P_{n},Q_{n}$ of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §coordinates, let $\mathcal{P}_{2}(X)$ be the set of probability measures on $X$ with finite second moment and $W_{2}$ the quadratic Wasserstein distance on it. For $n\in\mathbb{N}$, let $\mathcal{P}_{2}(\mathbb{R}^{n})$ be the set of probability measures on $\mathbb{R}^{n}$ with finite second moment and $W_{2}^{(n)}$ the quadratic Wasserstein distance on it. Let $\mu,\nu\in\mathcal{P}_{2}(X)$ and $n\in\mathbb{N}$.

1. (Projected measures)¶ $(p_{n})_{\#}\mu\in\mathcal{P}_{2}(\mathbb{R}^{n})$ and $(P_{n})_{\#}\mu\in\mathcal{P}_{2}(X)$.

2. (Isometry)¶ $W_{2}\bigl((P_{n})_{\#}\mu,(P_{n})_{\#}\nu\bigr)=W_{2}^{(n)}\bigl((p_{n})_{\#}\mu,(p_{n})_{\#}\nu\bigr)$.

3. (Monotonicity)¶ $W_{2}^{(n)}\bigl((p_{n})_{\#}\mu,(p_{n})_{\#}\nu\bigr)\le W_{2}^{(n+1)}\bigl((p_{n+1})_{\#}\mu,(p_{n+1})_{\#}\nu\bigr)\le W_{2}(\mu,\nu)$.

4. (Tails)¶ $W_{2}\bigl((P_{n})_{\#}\mu,\mu\bigr)^{2}\le\int_{X}|Q_{n}x|^{2}\,\mu(dx)$, and the real sequence $\bigl(\int_{X}|Q_{m}x|^{2}\,\mu(dx)\bigr)_{m\in\mathbb{N}}$ has limit $0$.

5. (Limit)¶ The real sequence $\bigl(W_{2}^{(m)}((p_{m})_{\#}\mu,(p_{m})_{\#}\nu)\bigr)_{m\in\mathbb{N}}$ has limit $W_{2}(\mu,\nu)$.