Proof of Completeness of the Quadratic Wasserstein Space over Euclidean Space

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

Throughout, $\mathcal{P}(\mathbb{R}^{d})$, $M_{2}$, $\mathcal{P}_{2}(\mathbb{R}^{d})$, the couplings $\Pi(\cdot,\cdot)$, the quadratic cost $I$ and $W_{2}$ are those of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation §dimensions with $m=d$, that is, of 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, Couplings of Two Probability Measures on Euclidean Space and Their Quadratic Cost §coupling, Couplings of Two Probability Measures on Euclidean Space and Their Quadratic Cost §cost and The Quadratic Wasserstein Distance on Euclidean Space §distance, and integrals against elements of $\mathcal{P}(\mathbb{R}^{q})$ are those of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §measures. The map $x\mapsto\lVert x\rVert$ on $\mathbb{R}^{d}$ is Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §functions, so for every real $K$ the set $\{x:\,K<\lVert x\rVert\}$ belongs to $\mathcal{B}(\mathbb{R}^{d})$ by the description of measurability in Measure Spaces and the Lebesgue Integral: Standing Notation §measurable; its indicator is measurable by claim 1 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, and the product $x\mapsto\lVert x\rVert^{2}\mathbf{1}_{\{K<\lVert x\rVert\}}(x)$ of it with the Borel map $x\mapsto\lVert x\rVert^{2}$ of Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §functions is a nonnegative Borel function by claim 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions. For $\rho\in\mathcal{P}(\mathbb{R}^{d})$, a set $A\in\mathcal{B}(\mathbb{R}^{d})$ and a nonnegative Borel function $f$ on $\mathbb{R}^{d}$ we use the convention $\int_{A}f\,d\rho:=\int_{\mathbb{R}^{d}}f\,\mathbf{1}_{A}\,d\rho$. For $\rho\in\mathcal{P}(\mathbb{R}^{d})$ and real $K$ we write

$$T_{K}(\rho)=\int_{\{x:\,K<\lVert x\rVert\}}\lVert x\rVert^{2}\,\rho(dx)=\int_{\mathbb{R}^{d}}\lVert x\rVert^{2}\mathbf{1}_{\{K<\lVert x\rVert\}}(x)\,\rho(dx)\in[0,\infty],$$

which is exactly the integral appearing in Wasserstein Convergence from Weak Convergence with Uniformly Integrable Second Moments, and Wasserstein Compactness under a Superquadratic Moment Bound §convergence. Since $0\le\lVert x\rVert^{2}\mathbf{1}_{\{K<\lVert x\rVert\}}(x)\le\lVert x\rVert^{2}$, monotonicity of the integral (claim 1 of Linearity and Monotonicity of the Lebesgue Integral) gives $T_{K}(\rho)\le M_{2}(\rho)$, so $T_{K}(\rho)$ is a nonnegative real number when $\rho\in\mathcal{P}_{2}(\mathbb{R}^{d})$. Moreover, if $K\le K'$ then $\mathbf{1}_{\{K'<\lVert x\rVert\}}\le\mathbf{1}_{\{K<\lVert x\rVert\}}$ pointwise, so again by claim 1 of Linearity and Monotonicity of the Lebesgue Integral

$$T_{K'}(\rho)\le T_{K}(\rho)\qquad\text{whenever }K\le K'.\tag{1}$$

Let $(\mu_{n})_{n\in\mathbb{N}}$ be a Cauchy sequence in the metric space $(\mathcal{P}_{2}(\mathbb{R}^{d}),W_{2})$ of The Quadratic Wasserstein Distance is a Metric on the Wasserstein Space §metric. By Complete Metric Space we must produce $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ to which $(\mu_{n})$ converges.

Step 1 (Tails of a single measure). Let $\rho\in\mathcal{P}_{2}(\mathbb{R}^{d})$ and let $\eta$ be a positive real number. We show that there is a positive real $K$ with $T_{K}(\rho)<\eta$. For $j\in\mathbb{N}$ put $f_{j}(x)=\lVert x\rVert^{2}\mathbf{1}_{\{j<\lVert x\rVert\}}(x)$, a measurable real-valued function as shown above, and put $g(x)=\lVert x\rVert^{2}$. Then $|f_{j}(x)|\le g(x)$ for all $x$ and $j$, and $g$ is integrable with respect to $\rho$ by the criterion of Measure Spaces and the Lebesgue Integral: Standing Notation §integral, since $\int_{\mathbb{R}^{d}}|g|\,d\rho=M_{2}(\rho)<\infty$. For fixed $x\in\mathbb{R}^{d}$, the Archimedean property of the reals gives $j_{0}\in\mathbb{N}$ with $\lVert x\rVert<j_{0}$, and then $f_{j}(x)=0$ for every $j\ge j_{0}$; so $(f_{j}(x))_{j}$ converges to $0$. By claim 3 of Dominated Convergence Theorem, applied on the measure space $(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}),\rho)$ with limit function $0$, the real sequence $(\int f_{j}\,d\rho)_{j}$ converges to $0$; here each $f_{j}$ is nonnegative, so its negative part $f_{j}^{-}$ is $0$ and its integral as an integrable function equals its integral as a nonnegative function, namely $T_{j}(\rho)\ge0$, so the sequence is $(T_{j}(\rho))_{j\in\mathbb{N}}$. By Limit of a Sequence of Real Numbers there is $j_{1}\in\mathbb{N}$ with $T_{j}(\rho)<\eta$ for all $j\ge j_{1}$; then $K=j_{1}+1$ is a natural number, hence a positive real by The Real Numbers: Standing Notation and Background §numbers, and $T_{K}(\rho)<\eta$.

Step 2 (Bounded second moments). Applying Cauchy Sequence in a Metric Space with $\varepsilon=1$ gives $N_{1}\in\mathbb{N}$ with $W_{2}(\mu_{n},\mu_{l})<1$ for all $n,l\ge N_{1}$. For $n\ge N_{1}$ we have $W_{2}(\mu_{N_{1}},\mu_{n})\le1$, so The Coordinate Fields of a Coupling, and the Second Moment as a Lipschitz Function of the Wasserstein Distance §ball, with $\mu_{N_{1}}$ in place of $\mu$, $\mu_{n}$ in place of $\nu$ and $R=1$, gives $M_{2}(\mu_{n})\le(\sqrt{M_{2}(\mu_{N_{1}})}+1)^{2}$. Each $M_{2}(\mu_{n})$ is a nonnegative real number because $\mu_{n}\in\mathcal{P}_{2}(\mathbb{R}^{d})$. Let $B$ be the maximum of the real number $(\sqrt{M_{2}(\mu_{N_{1}})}+1)^{2}$ and of the finitely many real numbers $M_{2}(\mu_{n})$ with $n\in\mathbb{N}$ and $n<N_{1}$. Then $B$ is a nonnegative real number and

$$M_{2}(\mu_{n})\le B\qquad\text{for every }n\in\mathbb{N}.\tag{2}$$

Step 3 (A weakly convergent subsequence). By (2) and Tightness from Bounded Second Moments, and Tightness of the Couplings of Two Measures with Finite Second Moment §moment, applied with $m=d$, $\mathcal{M}=\{\mu_{n}:n\in\mathbb{N}\}\subseteq\mathcal{P}(\mathbb{R}^{d})$ and the bound $B$, the set $\{\mu_{n}:n\in\mathbb{N}\}$ is tight in $(\mathbb{R}^{d},d_{E})$; that is, by Tight Family of Borel Measures on a Metric Space §sequence, the sequence $(\mu_{n})$ is tight. By Prokhorov's Theorem on Euclidean Space: a Tight Sequence of Probability Measures Has a Weakly Convergent Subsequence there are a strictly increasing sequence $(n_{j})_{j\in\mathbb{N}}$ in $\mathbb{N}$ and a measure $\mu\in\mathcal{P}(\mathbb{R}^{d})$ such that $(\mu_{n_{j}})_{j\in\mathbb{N}}$ converges weakly to $\mu$ on $(\mathbb{R}^{d},d_{E})$, in the sense of Weak Convergence of Finite Borel Measures on a Metric Space.

Step 4 (A pointwise inequality). Let $K$ and $R$ be nonnegative reals and let $x,y\in\mathbb{R}^{d}$. We claim

$$\lVert x\rVert^{2}\mathbf{1}_{\{K<\lVert x\rVert\}}(x)\le2\lVert x-y\rVert^{2}+2\lVert y\rVert^{2}\mathbf{1}_{\{R<\lVert y\rVert\}}(y)+2R^{2}\mathbf{1}_{\{K<\lVert x\rVert\}}(x).\tag{3}$$

If $\lVert x\rVert\le K$ the left side is $0$ and the right side is nonnegative. If $K<\lVert x\rVert$, then by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §functions we have $\lVert x\rVert^{2}\le2\lVert y\rVert^{2}+2\lVert x-y\rVert^{2}$; and $\lVert y\rVert^{2}\le\lVert y\rVert^{2}\mathbf{1}_{\{R<\lVert y\rVert\}}(y)+R^{2}$, because when $R<\lVert y\rVert$ the first term on the right equals $\lVert y\rVert^{2}$, while when $\lVert y\rVert\le R$ we have $\lVert y\rVert^{2}\le R^{2}$ as both numbers are nonnegative. Combining the two inequalities gives (3).

Step 5 (Uniformly small tails along the whole sequence). We show: for every positive real $\eta$ there is a positive real $K$ with $T_{K}(\mu_{n})<\eta$ for every $n\in\mathbb{N}$. Fix $\eta>0$; the remaining quantities are chosen in the following order.

First, apply Cauchy Sequence in a Metric Space with $\varepsilon=\sqrt{\eta/6}$, a positive real, to get $N\in\mathbb{N}$ with $W_{2}(\mu_{n},\mu_{l})<\sqrt{\eta/6}$ for all $n,l\ge N$; in particular $W_{2}(\mu_{n},\mu_{N})^{2}<\eta/6$ for every $n\ge N$. Second, apply Step 1 to $\rho=\mu_{N}$ and $\eta/6$ to get a positive real $R$ with $T_{R}(\mu_{N})<\eta/6$. Third, with $B$ from Step 2, put $K_{0}=1+6R^{2}B/\eta$, a real number with $K_{0}\ge1$. Fourth, for each of the finitely many $n\in\mathbb{N}$ with $n<N$ apply Step 1 to $\rho=\mu_{n}$ and $\eta$ to get a positive real $K_{n}$ with $T_{K_{n}}(\mu_{n})<\eta$. Finally let $K$ be the maximum of $K_{0}$ and of the numbers $K_{n}$ for $n<N$ (just $K_{0}$ if there is no such $n$); $K$ is positive.

For $n<N$ we have $K_{n}\le K$, so $T_{K}(\mu_{n})\le T_{K_{n}}(\mu_{n})<\eta$ by (1).

Let now $n\ge N$. By Existence of an Optimal Coupling of Two Probability Measures with Finite Second Moment, applied with $m=d$ to $\mu_{n},\mu_{N}\in\mathcal{P}_{2}(\mathbb{R}^{d})$, there is $\pi_{n}\in\Pi(\mu_{n},\mu_{N})$ with $I(\pi_{n})=W_{2}(\mu_{n},\mu_{N})^{2}$. Write $\mathrm{pr}_{1},\mathrm{pr}_{2}:\mathbb{R}^{d+d}\to\mathbb{R}^{d}$ for the coordinate projections of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pairs, which are Borel. Apply (3) with $x=\mathrm{pr}_{1}(z)$ and $y=\mathrm{pr}_{2}(z)$ for each $z\in\mathbb{R}^{d+d}$; every term of (3) is then a nonnegative Borel function of $z$, as a composition of the Borel functions above with Borel projections (Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps), the term $z\mapsto\lVert\mathrm{pr}_{1}(z)-\mathrm{pr}_{2}(z)\rVert^{2}$ being Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §functions (claim 4 there, with $q=d$). Integrating against $\pi_{n}$ and using linearity and monotonicity for nonnegative functions (claim 1 of Linearity and Monotonicity of the Lebesgue Integral) gives

$$\int\lVert\mathrm{pr}_{1}(z)\rVert^{2}\mathbf{1}_{\{K<\lVert\mathrm{pr}_{1}(z)\rVert\}}\,\pi_{n}(dz)\le2I(\pi_{n})+2\int\lVert\mathrm{pr}_{2}(z)\rVert^{2}\mathbf{1}_{\{R<\lVert\mathrm{pr}_{2}(z)\rVert\}}\,\pi_{n}(dz)+2R^{2}\int\mathbf{1}_{\{K<\lVert\mathrm{pr}_{1}(z)\rVert\}}\,\pi_{n}(dz),$$

where the first term on the right is identified with the quadratic cost by Couplings of Two Probability Measures on Euclidean Space and Their Quadratic Cost §cost. Since $(\mathrm{pr}_{1})_{\#}\pi_{n}=\mu_{n}$ and $(\mathrm{pr}_{2})_{\#}\pi_{n}=\mu_{N}$ by Couplings of Two Probability Measures on Euclidean Space and Their Quadratic Cost §coupling, the change-of-variables formula of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward turns the left side into $T_{K}(\mu_{n})$, the middle integral into $T_{R}(\mu_{N})$, and the last integral into $\int_{\mathbb{R}^{d}}\mathbf{1}_{\{K<\lVert x\rVert\}}\,d\mu_{n}=\mu_{n}(\{x:\,K<\lVert x\rVert\})$, the last equality by The Integral of an Indicator Function is the Measure of the Set. Thus

$$T_{K}(\mu_{n})\le2W_{2}(\mu_{n},\mu_{N})^{2}+2T_{R}(\mu_{N})+2R^{2}\mu_{n}(\{x:\,K<\lVert x\rVert\}).\tag{4}$$

To bound the last term, note that $K^{2}\mathbf{1}_{\{K<\lVert x\rVert\}}(x)\le\lVert x\rVert^{2}$ for every $x$, so by claim 1 of Linearity and Monotonicity of the Lebesgue Integral, The Integral of an Indicator Function is the Measure of the Set and (2), $K^{2}\mu_{n}(\{x:\,K<\lVert x\rVert\})\le M_{2}(\mu_{n})\le B$. Since $K\ge K_{0}\ge1$ we have $K^{2}\ge K\ge K_{0}>6R^{2}B/\eta$, the last inequality being strict because $K_{0}=1+6R^{2}B/\eta$; as $K^{2}>0$ and $R^{2}B\ge0$ (recall $B\ge0$), this gives $\mu_{n}(\{x:\,K<\lVert x\rVert\})\le B/K^{2}$ and $6R^{2}B<\eta K^{2}$, hence $2R^{2}\mu_{n}(\{x:\,K<\lVert x\rVert\})\le2R^{2}B/K^{2}<\eta/3$ (the strict inequality holding also when $R^{2}B=0$). The first two terms of (4) are below $2\cdot\eta/6=\eta/3$ each by the first and second choices. Hence $T_{K}(\mu_{n})<\eta$ for every $n\ge N$, and so for every $n\in\mathbb{N}$.

Step 6 (Wasserstein convergence of the subsequence). Step 5 holds for every $n\in\mathbb{N}$, in particular for every $n_{j}$: for every positive real $\eta$ there is a positive real $K$ with $T_{K}(\mu_{n_{j}})<\eta$ for every $j\in\mathbb{N}$. Together with $\mu_{n_{j}}\in\mathcal{P}_{2}(\mathbb{R}^{d})$, $\mu\in\mathcal{P}(\mathbb{R}^{d})$ and the weak convergence of Step 3, these are exactly the hypotheses of Wasserstein Convergence from Weak Convergence with Uniformly Integrable Second Moments, and Wasserstein Compactness under a Superquadratic Moment Bound §convergence, applied with $m=d$ to the sequence $(\mu_{n_{j}})_{j\in\mathbb{N}}$. Therefore $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ and the real sequence $(W_{2}(\mu_{n_{j}},\mu))_{j\in\mathbb{N}}$ has limit $0$.

Step 7 (The whole sequence converges). Let $\varepsilon$ be a positive real. By Cauchy Sequence in a Metric Space choose $N'\in\mathbb{N}$ with $W_{2}(\mu_{n},\mu_{l})<\varepsilon/2$ for all $n,l\ge N'$. By Step 6 and Limit of a Sequence of Real Numbers choose $J\in\mathbb{N}$ with $W_{2}(\mu_{n_{j}},\mu)<\varepsilon/2$ for all $j\ge J$. Let $j$ be the larger of $N'$ and $J$. By Strictly Increasing Sequences of Natural Numbers Dominate Their Index, $j\le n_{j}$, so $n_{j}\ge N'$. For every $n\ge N'$, the triangle inequality The Quadratic Wasserstein Distance is a Metric on the Wasserstein Space §triangle gives

$$W_{2}(\mu_{n},\mu)\le W_{2}(\mu_{n},\mu_{n_{j}})+W_{2}(\mu_{n_{j}},\mu)<\varepsilon/2+\varepsilon/2=\varepsilon .$$

Since $\varepsilon$ was arbitrary, $(\mu_{n})$ converges to $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ in $(\mathcal{P}_{2}(\mathbb{R}^{d}),W_{2})$ in the sense of Convergent Sequence in a Metric Space. As the Cauchy sequence $(\mu_{n})$ was arbitrary, $(\mathcal{P}_{2}(\mathbb{R}^{d}),W_{2})$ is a complete metric space by Complete Metric Space, which proves (Completeness).