The Optimal Maps of a Uniquely Mapped Pair and of Its Reverse are Mutually Inverse Almost Everywhere

In the setting of Probability Measures on Euclidean Space and Random Vectors: Standing Notation, let $d\in\mathbb{N}$ satisfy $1\le d$ and let $\mu,\nu$ belong to the set $\mathcal{P}_{2}(\mathbb{R}^{d})$ of probability measures with finite second moment. Suppose that the ordered pair $(\mu,\nu)$ is uniquely mapped and that so is $(\nu,\mu)$, let $T$ be an optimal map from $\mu$ to $\nu$, and let $T'$ be an optimal map from $\nu$ to $\mu$.

The composites $T'\circ T$ and $T\circ T'$ are Borel. For Borel maps $S,S':\mathbb{R}^{d}\to\mathbb{R}^{d}$ the function $x\mapsto\lVert S(x)-S'(x)\rVert^{2}$ 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 vanishes exactly where $S$ and $S'$ agree, 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 together with the metric axioms of $\mathbb{R}^n$ with the Euclidean Distance Is a Complete Metric Space with Coordinatewise Convergence and claims 1 and 3 of Zero Products and Elementary Identities in a Field; hence the two sets appearing below, preimages of the Borel set $\mathbb{R}\setminus\{0\}$ under such a function with $S'=\mathrm{id}$, belong to $\mathcal{B}(\mathbb{R}^{d})$.

1. (Mutually inverse almost everywhere)(i)

$$\mu\bigl(\{x\in\mathbb{R}^{d}:\ T'(T(x))\ne x\}\bigr)=0,\qquad \nu\bigl(\{y\in\mathbb{R}^{d}:\ T(T'(y))\ne y\}\bigr)=0 .$$