Continuity of the Coordinate Projections and of the Quadratic Cost Function, and Passage of a Cost Bound to a Weak Limit

In the setting of Probability Measures on Euclidean Space and Random Vectors: Standing Notation, let $m\in\mathbb{N}$ satisfy $1\le m$ and let $d_{E}$ denote the Euclidean distance, on $\mathbb{R}^{m}$ and on $\mathbb{R}^{m+m}$ as the dimension of its arguments requires, so that $d_{E}(x,y)=\lVert x-y\rVert$ by claim 2 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square. Let $\mathbb{R}$ carry the absolute-value metric, and let continuity and the Lipschitz property be those of the metric spaces so named. As in Probability Measures on Euclidean Space and Random Vectors: Standing Notation (v), $\mathrm{pr}_{1},\mathrm{pr}_{2}:\mathbb{R}^{m+m}\to\mathbb{R}^{m}$ are the coordinate projections for the splitting $m+m$.

Let $\varphi:\mathbb{R}^{m+m}\to\mathbb{R}$ be the map with $\varphi(z)=\lVert\mathrm{pr}_{1}(z)-\mathrm{pr}_{2}(z)\rVert^{2}$, 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 nonnegative by Nonnegativity of Squares in an Ordered Field, being a square, and for $\pi\in\mathcal{P}(\mathbb{R}^{m+m})$ put

$$J(\pi)=\int_{\mathbb{R}^{m+m}}\varphi\,d\pi\in[0,\infty],$$

the integral of Probability Measures on Euclidean Space and Random Vectors: Standing Notation (iii) of $\varphi$ read as a map into $[0,\infty]$. For $\mu,\nu\in\mathcal{P}(\mathbb{R}^{m})$ and a coupling $\pi\in\Pi(\mu,\nu)$, $J(\pi)$ is the quadratic cost $I(\pi)$.

Then the following hold.

1. (The coordinate projections are nonexpansive)(i) Let $q,p\in\mathbb{N}$ satisfy $1\le q$ and $1\le p$. Then the coordinate projections $\mathrm{pr}^{q,p}_{1}:\mathbb{R}^{q+p}\to\mathbb{R}^{q}$ and $\mathrm{pr}^{q,p}_{2}:\mathbb{R}^{q+p}\to\mathbb{R}^{p}$ of Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (i) are Lipschitz with constant $1$, and continuous on $\mathbb{R}^{q+p}$.

2. (The cost function is continuous)(ii) The map $\mathbb{R}^{m+m}\to\mathbb{R}^{m}$ with value $\mathrm{pr}_{1}(z)-\mathrm{pr}_{2}(z)$ at $z$ is Lipschitz with constant $2$, and $\varphi$ is continuous on $\mathbb{R}^{m+m}$.

3. (A cost bound passes to a weak limit)(iii) Let $(\pi_{n})_{n\in\mathbb{N}}$ be a sequence in $\mathcal{P}(\mathbb{R}^{m+m})$ that converges weakly to $\pi\in\mathcal{P}(\mathbb{R}^{m+m})$ on the metric space $(\mathbb{R}^{m+m},d_{E})$, and let $c\in\mathbb{R}$ be such that for every $\varepsilon\in\mathbb{R}$ with $0<\varepsilon$ there is $N\in\mathbb{N}$ with

$$J(\pi_{n})\le c+\varepsilon\qquad\text{for every }n\in\mathbb{N}\text{ with }N\le n .$$

Then $J(\pi)\le c$; in particular $J(\pi)$ is a real number.