Convergence in the Wasserstein Distance Implies Weak Convergence and Convergence of Integrals of Continuous Functions of Quadratic Growth

In the setting of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation, let $m\in\mathbb{N}$ satisfy $1\le m$, with the Wasserstein space $(\mathcal{P}_{2}(\mathbb{R}^{m}),W_{2})$. Weak convergence $\mu_{n}\Rightarrow\mu$ is that of that definition on $\mathbb{R}^{m}$ with the Euclidean distance; limits of real sequences are those of that definition; continuity of a function $\mathbb{R}^{m}\to\mathbb{R}$ refers to the Euclidean distance and the metric of The Absolute Value Metric on the Real Line; and integrable is as in Probability Measures on Euclidean Space and Random Vectors: Standing Notation §measures. Let $\mu\in\mathcal{P}_{2}(\mathbb{R}^{m})$ and let $\mu_{n}\in\mathcal{P}_{2}(\mathbb{R}^{m})$ for $n\in\mathbb{N}$ satisfy $\lim_{n\to\infty}W_{2}(\mu_{n},\mu)=0$.

1. (Weak convergence)¶ $\mu_{n}\Rightarrow\mu$.

2. (Quadratic growth)¶ Let $h:\mathbb{R}^{m}\to\mathbb{R}$ be continuous, and let $A\in\mathbb{R}$ satisfy $|h(x)|\le A\bigl(1+\lVert x\rVert^{2}\bigr)$ for every $x\in\mathbb{R}^{m}$. Then $h$ is integrable with respect to $\mu$ and to every $\mu_{n}$, and