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

In the setting of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation, having the map property is the property of a subset of $\mathcal{P}_{2}(\mathbb{R}^{d})$ defined there. Then the following hold.

1. (Absolutely continuous measures, in every dimension)(i) Let $Q\subseteq\mathcal{P}_{2}(\mathbb{R}^{d})$ and suppose that every $\mu\in Q$ is absolutely continuous. Then $Q$ has the map property.

2. (Atomless measures on the line)(ii) Let $d=1$, with the identification of $\mathbb{R}$ and $\mathbb{R}^{1}$ recorded in On the Real Line an Atomless Source is Uniquely Mapped, by a Nondecreasing Optimal Map, let $Q\subseteq\mathcal{P}_{2}(\mathbb{R})$ and suppose that every $\mu\in Q$ is atomless. Then $Q$ has the map property.