Convergence Along Couplings of Measures Carrying Square-Integrable Vector Fields on the Torus

In the setting of Optimal Transport on the Flat Torus: Standing Notation, let $\mu\in\mathcal{P}(\mathbb{T}^{d})$ and $\eta\in L^{2}(\mu;\mathbb{R}^{d})$, and for $n\in\mathbb{N}$ let $\mu_{n}\in\mathcal{P}(\mathbb{T}^{d})$ and $\eta_{n}\in L^{2}(\mu_{n};\mathbb{R}^{d})$. For $\gamma\in\Pi(\mu_{n},\mu)$ and Borel representatives of $\eta_{n}$ and $\eta$, the function $w\mapsto\lVert\eta_{n}(\mathrm{pr}_{1}(w))-\eta(\mathrm{pr}_{2}(w))\rVert^{2}$ on $\mathbb{R}^{d+d}$ is Borel and nonnegative, and its integral with respect to $\gamma$ is at most $2\lVert\eta_{n}\rVert_{\mu_{n}}^{2}+2\lVert\eta\rVert_{\mu}^{2}$, by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §functions and the change-of-variables formula; that integral does not depend on the representatives, by The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §comparison, and is written $D_{\gamma}(\eta_{n},\eta)$.

¶The sequence $((\mu_{n},\eta_{n}))_{n\in\mathbb{N}}$ converges along couplings to $(\mu,\eta)$ if there are $\gamma_{n}\in\Pi(\mu_{n},\mu)$, for $n\in\mathbb{N}$, such that the real sequences $(I_{\mathbb{T}}(\gamma_{n}))_{n\in\mathbb{N}}$ and $(D_{\gamma_{n}}(\eta_{n},\eta))_{n\in\mathbb{N}}$ both converge to $0$.