Existence of an Optimal Coupling of Two Probability Measures with Finite Second Moment

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 $\mu,\nu$ belong to the set $\mathcal{P}_{2}(\mathbb{R}^{m})$ of probability measures with finite second moment. Let $\Pi(\mu,\nu)$ be the set of their couplings, $I$ the quadratic cost, and $W_{2}(\mu,\nu)$ their quadratic Wasserstein distance.

Then there exists $\pi\in\Pi(\mu,\nu)$ with

$$I(\pi)=W_{2}(\mu,\nu)^{2},$$

that is, an optimal coupling of $\mu$ and $\nu$; and every such $\pi$ satisfies $I(\pi)\le I(\sigma)$ for every $\sigma\in\Pi(\mu,\nu)$.