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

In the setting of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation, with the coordinate maps $p_{n}:X\to\mathbb{R}^{n}$ and the maps $Q_{n}=\mathrm{id}_{X}-P_{n}$ 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 The Second Moment of a Borel Probability Measure on a Hilbert Space and the Probability Measures with Finite Second Moment §space and $W_{2}$ the quadratic Wasserstein distance on it. Sequences and limits of sequences of real numbers are those of those definitions. For $n\in\mathbb{N}$ the function $x\mapsto|Q_{n}x|^{2}$ is continuous, hence Borel, and satisfies $|Q_{n}x|^{2}\le|x|^{2}$, by Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §continuity; so it is integrable with respect to every member of $\mathcal{P}_{2}(X)$, by The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §comparison and The Second Moment of a Borel Probability Measure on a Hilbert Space and the Probability Measures with Finite Second Moment §space. Let $\nu\in\mathcal{P}_{2}(X)$ and let $(\lambda_{n})_{n\in\mathbb{N}}$ be a sequence in $\mathcal{P}_{2}(X)$ such that $(p_{n})_{\#}\lambda_{n}=(p_{n})_{\#}\nu$ for every $n\in\mathbb{N}$ and

$$\lim_{n\to\infty}\int_{X}|Q_{n}x|^{2}\,\lambda_{n}(dx)=0 .$$

¶ Then $\lim_{n\to\infty}W_{2}(\lambda_{n},\nu)=0$.