Proof of Mean-Square Stability of the Optimal Map on the Space of Square-Integrable Random Vectors

Throughout, each result cited is universally quantified over the data appearing in its own statement and is applied to the data named here. Being an optimal map, $T$ is Borel and satisfies $T_{\#}\mu=\nu$.

Step 1 (Claim 1). Let $X\in L^{2}(\Omega;\mathbb{R}^{d})$ with $\mathcal{L}(X)=\mu$ and let $X$ also denote a representative of the class, a random vector on $(\Omega,\mathcal{F},P)$. The composition $T\circ X$ is measurable with respect to $\mathcal{F}$ and $\mathcal{B}(\mathbb{R}^{d})$ by claim 4 of Borel Measurability and Bounded Integration on a Metric Space, hence is a random vector.

Its law is $\nu$: for $B\in\mathcal{B}(\mathbb{R}^{d})$ the set $T^{-1}(B)$ is Borel, so by the definition of the law and Probability Measures on Euclidean Space and Random Vectors: Standing Notation (iv),

$$\mathcal{L}(T\circ X)(B)=P\bigl(T\circ X\in B\bigr)=P\bigl(X\in T^{-1}(B)\bigr)=\mu\bigl(T^{-1}(B)\bigr)=T_{\#}\mu(B)=\nu(B).$$

By Random Vector and Its Law (iii) the law $\mathcal{L}(X)$ is given by $\mathcal{L}(X)(B)=P(X^{-1}(B))$, so it is the image measure of $P$ under $X$ in the sense of claim 1 of Image Measures and Measures with a Density: Measures, Integrals and Change of Variables, applied to the measure space $(\Omega,\mathcal{F},P)$ and the measurable map $X$. Hence, by claim 2 of that lemma applied to the nonnegative function $y\mapsto\lVert T(y)\rVert^{2}$, which is Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (iv), and by Probability Measures on Euclidean Space and Random Vectors: Standing Notation (vi), under which $\mathbb{E}[V]=\int_{\Omega}V\,dP$,

$$\mathbb{E}\bigl[\lVert T\circ X\rVert^{2}\bigr]=\int_{\mathbb{R}^{d}}\lVert T\rVert^{2}\,d\mu=\int_{\mathbb{R}^{d}}\lVert y\rVert^{2}\,(T_{\#}\mu)(dy)=M_{2}(\nu)<\infty ,$$

the middle identity again by the change-of-variables formula of Probability Measures on Euclidean Space and Random Vectors: Standing Notation (iv) and the last by The Second Moment of a Probability Measure on Euclidean Space and the Probability Measures with Finite Second Moment (i), $\nu$ having finite second moment. Hence $T\circ X$ is a square-integrable random vector and $\lVert T\circ X\rVert_{L^{2}}^{2}=M_{2}(\nu)$ by The Space of Square-Integrable Random Vectors (iii).

If $X'$ is another representative of the class of $X$, then $P(X=X')=1$, and $\{\omega:T(X(\omega))\ne T(X'(\omega))\}\subseteq\{\omega:X(\omega)\ne X'(\omega)\}$, a set of probability $0$; by claim 1 of The Lebesgue Integral and Null Sets the former set is therefore contained in a null set, so $T\circ X\sim_{P}T\circ X'$ and the class of $T\circ X$ does not depend on the representative, by The Space of Square-Integrable Random Vectors (ii).

Step 2 (Claim 2). Let $(X_{n})$ and $(Y_{n})$ be as in claim 2. For each $n$ let $\pi_{n}=\mathcal{L}\bigl((X_{n},Y_{n})\bigr)$ be the joint law of the pair, as in The Joint Law of Two Classes of Square-Integrable Random Vectors: Marginals, Cost, and Invariance Under Shifts by a Vector Field. By the marginal and cost clauses of that lemma, $\pi_{n}\in\Pi(\mu,\nu)$ and

$$I(\pi_{n})=\lVert X_{n}-Y_{n}\rVert_{L^{2}}^{2}.$$

The sequence $(\lVert X_{n}-Y_{n}\rVert_{L^{2}})_{n}$ converges to $W_{2}(\mu,\nu)$ by hypothesis, so $(I(\pi_{n}))_{n}$ converges to $W_{2}(\mu,\nu)^{2}$ by claim 2 of Arithmetic of Limits of Real Sequences.

Therefore Mean-Square Stability of the Optimal Map of a Uniquely Mapped Pair Along Couplings of Nearly Optimal Cost (i) applies and gives that $\bigl(\int_{\mathbb{R}^{d+d}}D_{T}\,d\pi_{n}\bigr)_{n}$ converges to $0$, where $D_{T}(z)=\lVert T(\mathrm{pr}_{1}(z))-\mathrm{pr}_{2}(z)\rVert^{2}$.

Finally, applying claims 1 and 2 of Image Measures and Measures with a Density: Measures, Integrals and Change of Variables to the measure space $(\Omega,\mathcal{F},P)$, the measurable map $(X_{n},Y_{n}):\Omega\to\mathbb{R}^{d+d}$, whose image measure is $\pi_{n}$ by The Joint Law of Two Classes of Square-Integrable Random Vectors: Marginals, Cost, and Invariance Under Shifts by a Vector Field, and the nonnegative Borel function $D_{T}$, and using $\mathrm{pr}_{1}\bigl((X_{n},Y_{n})(\omega)\bigr)=X_{n}(\omega)$ and $\mathrm{pr}_{2}\bigl((X_{n},Y_{n})(\omega)\bigr)=Y_{n}(\omega)$ as recorded in Probability Measures on Euclidean Space and Random Vectors: Standing Notation (v),

$$\lVert T\circ X_{n}-Y_{n}\rVert_{L^{2}}^{2}=\mathbb{E}\bigl[\lVert T(X_{n})-Y_{n}\rVert^{2}\bigr]=\int_{\mathbb{R}^{d+d}}D_{T}\,d\pi_{n},$$

the first identity by The Space of Square-Integrable Random Vectors (iii). Hence $\bigl(\lVert T\circ X_{n}-Y_{n}\rVert_{L^{2}}^{2}\bigr)_{n}$ converges to $0$, and so does $\bigl(\lVert T\circ X_{n}-Y_{n}\rVert_{L^{2}}\bigr)_{n}$: given a positive real $\eta$, the inequality $\lVert T\circ X_{n}-Y_{n}\rVert_{L^{2}}^{2}\le\eta^{2}$ gives $\lVert T\circ X_{n}-Y_{n}\rVert_{L^{2}}\le\eta$ by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, both quantities being nonnegative.