The Torus Wasserstein Space is a Sequentially Compact Metric Space in which Optimal Couplings Exist

In the setting of The Flat Torus: Standing Notation, used with $n=d$ for a natural number $d$ with $1\le d$, and of Probability Measures on Euclidean Space and Random Vectors: Standing Notation, let $\mathcal{P}(\mathbb{T}^{d})$, the torus Wasserstein distance $W_{\mathbb{T}}$ and optimal couplings be as in Probability Measures on the Flat Torus, the Torus Cost of a Coupling, the Torus Wasserstein Distance and Optimal Couplings §measures. Then the following hold.

1. (Existence of optimal couplings)¶ For all $\mu,\nu\in\mathcal{P}(\mathbb{T}^{d})$ there is an optimal coupling of $\mu$ and $\nu$.

2. (Metric)¶ $W_{\mathbb{T}}$ is a metric on $\mathcal{P}(\mathbb{T}^{d})$.

3. (Sequential compactness)¶ Every sequence in $\mathcal{P}(\mathbb{T}^{d})$ has a subsequence that converges in the metric space $(\mathcal{P}(\mathbb{T}^{d}),W_{\mathbb{T}})$.