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

Each result cited below is universally quantified over the data in its own statement, and is applied to the data named at the point of use. The rules of Elementary Order Arithmetic in an Ordered Field and Elementary Arithmetic in an Ordered Field for adding, multiplying and comparing inequalities between real numbers, and the properties of the absolute value in Properties of the Absolute Value in an Ordered Field, are used without further mention. Integrals of nonnegative Borel functions are taken in $[0,\infty]$; linearity and monotonicity of integrals are claim 1 (nonnegative functions) and claim 2 (integrable functions, used only after integrability has been established) of Linearity and Monotonicity of the Lebesgue Integral. The Borel $\sigma$-algebra of the metric space $(\mathbb{R}^{m},d_{E})$ is $\mathcal{B}(\mathbb{R}^{m})$ by claim 5 of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets, and a continuous function $\mathbb{R}^{m}\to\mathbb{R}$ is Borel by claim 3 there.

Claim 1. For each $n$ let $\pi_{n}\in\Pi(\mu_{n},\mu)$ be an optimal coupling, so that $I(\pi_{n})=W_{2}(\mu_{n},\mu)^{2}$ (Existence of an Optimal Coupling of Two Probability Measures with Finite Second Moment, in dimension $m$). Let $f:\mathbb{R}^{m}\to\mathbb{R}$ be bounded and Lipschitz for the Euclidean distance and the absolute-value metric, with constant $L\ge0$; this is the notion of Lipschitz function of Couplings on Euclidean Space: Product Coupling, Swap, Finiteness of the Cost, Push-Forward Couplings, Modifying One Marginal, Quantisation, Gluing over a Finitely Supported Measure, and the Lipschitz Bound, as recorded there. By Couplings on Euclidean Space: Product Coupling, Swap, Finiteness of the Cost, Push-Forward Couplings, Modifying One Marginal, Quantisation, Gluing over a Finitely Supported Measure, and the Lipschitz Bound §lipschitz, $f$ is Borel and integrable with respect to $\mu_{n}$ and $\mu$, and

Since $(L\,W_{2}(\mu_{n},\mu))_{n}$ has limit $0$ (claim 3 of Arithmetic of Limits of Real Sequences), claim 3 of Order Properties of Limits of Real Sequences gives $\int f\,d\mu_{n}\to\int f\,d\mu$. As $\mathbb{R}^{m}$ is nonempty and $\mu_{n},\mu$ are probability measures on $(\mathbb{R}^{m},\mathcal{B}(\mathbb{R}^{m}))$, claim 1 of Portmanteau Theorem on a Metric Space gives $\mu_{n}\Rightarrow\mu$.

Claim 2. (i) The majorant. Since $0\le|h(0_{\mathbb{R}^{m}})|\le A$, $A\ge0$. The function $x\mapsto\lVert x\rVert$ is Lipschitz with constant $1$: by the triangle inequality (claim 6 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$), $\lVert x\rVert\le\lVert x-y\rVert+\lVert y\rVert$ and $\lVert y\rVert\le\lVert y-x\rVert+\lVert x\rVert$, and $\lVert y-x\rVert=\lVert x-y\rVert$ by homogeneity (claim 5 there, with the scalar $-1$), so $|\lVert x\rVert-\lVert y\rVert|\le\lVert x-y\rVert$ by claim 6 of Properties of the Absolute Value in an Ordered Field; hence it is continuous (A Lipschitz Map is Uniformly Continuous); so $G(x)=A(1+\lVert x\rVert\,\lVert x\rVert)$ is continuous by claims 1, 2 and 3 of Continuity of Sums and Products of Real-Valued Functions on a Metric Space, and Borel. For $\nu\in\mathcal{P}_{2}(\mathbb{R}^{m})$, $\int\lVert x\rVert^{2}\,\nu(dx)=M_{2}(\nu)<\infty$ (The Second Moment of a Probability Measure on Euclidean Space and the Probability Measures with Finite Second Moment §moment, The Second Moment of a Probability Measure on Euclidean Space and the Probability Measures with Finite Second Moment §space), so $\int G\,d\nu=A(1+M_{2}(\nu))<\infty$, $\nu$ being a probability measure. As $h$ is continuous, hence Borel, and $|h|\le G$, $\int|h|\,d\nu\le\int G\,d\nu<\infty$, so $h$ is integrable with respect to $\nu$; this applies to $\nu=\mu$ and to every $\nu=\mu_{n}$.

(ii) Convergence of the majorant. By The Coordinate Fields of a Coupling, and the Second Moment as a Lipschitz Function of the Wasserstein Distance §lipschitz, $|\sqrt{M_{2}(\mu_{n})}-\sqrt{M_{2}(\mu)}|\le W_{2}(\mu_{n},\mu)$, so $\sqrt{M_{2}(\mu_{n})}\to\sqrt{M_{2}(\mu)}$ by claim 3 of Order Properties of Limits of Real Sequences, and, squaring (claim 2 of Arithmetic of Limits of Real Sequences and Existence and Uniqueness of the Nonnegative Square Root), $M_{2}(\mu_{n})\to M_{2}(\mu)$. By claims 1 and 3 of Arithmetic of Limits of Real Sequences, $\int G\,d\mu_{n}\to\int G\,d\mu$.

(iii) A lower bound. Let $g:\mathbb{R}^{m}\to\mathbb{R}$ be continuous and nonnegative, and integrable with respect to $\mu$. We show: for every positive $\varepsilon$ there is $N\in\mathbb{N}$ with $\int g\,d\mu-\varepsilon<\int g\,d\mu_{n}$ for every $n\ge N$ with $g$ integrable with respect to $\mu_{n}$. For $k\in\mathbb{N}$, read in $\mathbb{R}$, let $g_{k}=\min\{g,k\}$; it is continuous by claim 4 of Continuity of the Absolute Value, Maximum and Minimum of Real-Valued Functions on a Metric Space, with $0\le g_{k}\le k$, so bounded and Borel, and by Claim 1 and Weak Convergence of Finite Borel Measures on a Metric Space, $\int g_{k}\,d\mu_{n}\to\int g_{k}\,d\mu$. The functions $g_{k}$ are nondecreasing in $k$ and $g_{k}(x)=g(x)$ once $k\ge g(x)$ (claim 1 of The Archimedean Property of the Real Numbers), so Monotone Convergence Theorem gives $\int g\,d\mu=\sup_{k}\int g_{k}\,d\mu$. Given $\varepsilon>0$, choose first $k$ with $\int g_{k}\,d\mu>\int g\,d\mu-\varepsilon/2$ (Approximation Property of the Supremum and the Infimum in $\mathbb{R}$), then $N$ with $|\int g_{k}\,d\mu_{n}-\int g_{k}\,d\mu|<\varepsilon/2$ for $n\ge N$. For such $n$, $\int g\,d\mu_{n}\ge\int g_{k}\,d\mu_{n}>\int g\,d\mu-\varepsilon$, because $g_{k}\le g$.

(iv) Conclusion. The functions $g_{+}=G-h$ and $g_{-}=G+h$ are continuous (claims 2 and 4 of Continuity of Sums and Products of Real-Valued Functions on a Metric Space), nonnegative because $|h|\le G$, and integrable with respect to $\mu$ and every $\mu_{n}$ by (i). Let $\varepsilon>0$. By (iii) applied to $g_{+}$ and to $g_{-}$, and by (ii), there is $N$ such that for every $n\ge N$

For such $n$, by linearity of the integral,

So $|\int h\,d\mu_{n}-\int h\,d\mu|<2\varepsilon$ for $n\ge N$, and $\int h\,d\mu_{n}\to\int h\,d\mu$ by Limit of a Sequence of Real Numbers. $\blacksquare$