The Map Property of a Set of Probability Measures

In the setting of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation, let $Q$ be a subset of $\mathcal{P}_{2}(\mathbb{R}^{d})$. For $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ let $L^{2}(\mu;\mathbb{R}^{d})$ be the space of square-integrable vector fields against $\mu$ and $T_{\mu}\subseteq L^{2}(\mu;\mathbb{R}^{d})$ the tangent space at $\mu$, and let $\mathrm{id}$ be the identity map of $\mathbb{R}^{d}$, whose class lies in $L^{2}(\mu;\mathbb{R}^{d})$ by Basic Properties of the Tangent Space: Closed Subspace, the Identity Map Belongs to It, Second-Moment Limits, and Representation of Bounded Functionals on Gradients (ii).

(The map property)(i) The set $Q$ has the map property if for all $\mu\in Q$ and $\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ the ordered pair $(\mu,\nu)$ is uniquely mapped and, $T$ denoting an optimal map from $\mu$ to $\nu$, which exists by that clause and whose class in $L^{2}(\mu;\mathbb{R}^{d})$ is supplied by The Optimal Map as a Square-Integrable Vector Field: Integrability, Transport Cost and Uniqueness of the Class (i),

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

This condition does not depend on the choice of $T$: the pair being uniquely mapped, any two optimal maps from $\mu$ to $\nu$ have the same class in $L^{2}(\mu;\mathbb{R}^{d})$, and hence so do their displacements, by The Optimal Map as a Square-Integrable Vector Field: Integrability, Transport Cost and Uniqueness of the Class (iii).