The Torus Displacement Pairing of a Vector Field Along a Coupling

In the setting of Optimal Transport on the Flat Torus: Standing Notation, let $\mu,\nu\in\mathcal{P}(\mathbb{T}^{d})$, $\gamma\in\Pi(\mu,\nu)$, and $\eta\in L^{2}(\mu;\mathbb{R}^{d})$, the space of square-integrable vector fields against $\mu$. For a Borel representative $\eta$, the function $w\mapsto\eta(\mathrm{pr}_{1}(w))\cdot\varpi(\mathrm{pr}_{2}(w)-\mathrm{pr}_{1}(w))$ on $\mathbb{R}^{d+d}$ is Borel, by The Flat Torus Distance: Minimality of the Wrapped Displacement, Periodicity, the Metric on the Unit Cell and the Lipschitz Bound §lipschitz and Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §functions; its absolute value is at most $\tfrac{\sqrt d}{2}\lVert\eta(\mathrm{pr}_{1}(w))\rVert$ by The Flat Torus Distance: Minimality of the Wrapped Displacement, Periodicity, the Metric on the Unit Cell and the Lipschitz Bound §range and the Cauchy–Schwarz inequality Cauchy-Schwarz Inequality for the Euclidean Dot Product, and $\int\lVert\eta\circ\mathrm{pr}_{1}\rVert\,d\gamma=\int\lVert\eta\rVert\,d\mu\le\lVert\eta\rVert_{\mu}$ by the change-of-variables formula and Cauchy-Schwarz and Triangle Inequalities for the Mean-Square Norm; so it is integrable with respect to $\gamma$. Two representatives agree off a $\mu$-null Borel set $N$, so the two integrands agree off the $\gamma$-null set $\mathrm{pr}_{1}^{-1}(N)$ and have the same integral by The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §comparison.

(Torus displacement pairing)¶ The torus displacement pairing of $\eta$ along $\gamma$ is the real number

$$\mathcal{J}_{\mathbb{T}}(\eta,\gamma)=\int_{\mathbb{R}^{d+d}}\eta\bigl(\mathrm{pr}_{1}(w)\bigr)\cdot\varpi\bigl(\mathrm{pr}_{2}(w)-\mathrm{pr}_{1}(w)\bigr)\,\gamma(dw).$$