Proof of Two Sufficient Conditions for the Map Property: Absolute Continuity, and Atomlessness on the Line

Each result cited is universally quantified over the data in its own statement, and is applied here to the data named in the statement above. By The Map Property of a Set of Probability Measures (i) it suffices, in each claim, to fix $\mu\in Q$ and $\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ and to show that the ordered pair $(\mu,\nu)$ is uniquely mapped and that $\mathrm{id}-T\in T_{\mu}$ for an optimal map $T$ from $\mu$ to $\nu$; the choice of $T$ is immaterial, by the independence recorded in that clause.

Claim 1. Let $\mu\in Q$ and $\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$. By hypothesis $\mu$ is absolutely continuous, so Out of an Absolutely Continuous Measure the Optimal Map and the Optimal Displacement are Tangent (i) applies to the pair $(\mu,\nu)$: it is uniquely mapped, and there is an optimal map from $\mu$ to $\nu$. Let $T$ be such a map. By Out of an Absolutely Continuous Measure the Optimal Map and the Optimal Displacement are Tangent (ii) one has $\int_{\mathbb{R}^{d}}\lVert T\rVert^{2}\,d\mu=M_{2}(\nu)<\infty$, so that the class of $T$ belongs to $L^{2}(\mu;\mathbb{R}^{d})$, and

$$\mathrm{id}-T\in T_{\mu}.$$

The class of $\mathrm{id}-T$ named in that clause and the one named in The Map Property of a Set of Probability Measures (i), which is supplied by The Optimal Map as a Square-Integrable Vector Field: Integrability, Transport Cost and Uniqueness of the Class (i), are the same element of $L^{2}(\mu;\mathbb{R}^{d})$: both are the class of the Borel map $x\mapsto x-T(x)$, the class of a Borel map being determined by the map, by Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation (ii). Hence the condition of The Map Property of a Set of Probability Measures (i) holds for $\mu$ and $\nu$. As $\mu\in Q$ and $\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ were arbitrary, $Q$ has the map property.

Claim 2. Let $d=1$, let $\mu\in Q$ and let $\nu\in\mathcal{P}_{2}(\mathbb{R})$. By hypothesis $\mu$ is atomless, so On the Real Line an Atomless Source is Uniquely Mapped, by a Nondecreasing Optimal Map (ii) gives that the ordered pair $(\mu,\nu)$ is uniquely mapped. By Optimal Transport Maps and Uniquely Mapped Pairs of Probability Measures (ii) there is then an optimal map $T$ from $\mu$ to $\nu$; fix one. By The Optimal Map as a Square-Integrable Vector Field: Integrability, Transport Cost and Uniqueness of the Class (i), applied with $T$ in the role of the Borel map there and using $T_{\#}\mu=\nu$ from Optimal Transport Maps and Uniquely Mapped Pairs of Probability Measures (i), the classes of $T$ and of $\mathrm{id}-T$ belong to $L^{2}(\mu;\mathbb{R})$.

By On the Real Line the Tangent Space is the Whole Space of Square-Integrable Vector Fields (i), applied to $\mu\in\mathcal{P}_{2}(\mathbb{R})$, the tangent space at $\mu$ is the whole space:

$$T_{\mu}=L^{2}(\mu;\mathbb{R}).$$

Therefore $\mathrm{id}-T\in T_{\mu}$, and the condition of The Map Property of a Set of Probability Measures (i) holds for $\mu$ and $\nu$. As $\mu\in Q$ and $\nu\in\mathcal{P}_{2}(\mathbb{R})$ were arbitrary, $Q$ has the map property.