Proof of The Entropy Domain of a Diagonal Gaussian Reference Measure Has the Noise Map Property

Each result cited is universally quantified over the data in its own statement.

Claim 1. Let $\mu\in\mathcal{D}_{H}$. The data of Noise-Optimal Maps out of the Entropy Domain of a Diagonal Gaussian Reference Measure are present: the reference measure is $\rho=\gamma_{c}$, $\kappa$ is positive with $c_{k}\le\kappa\,a_{k}$ for every $k\in\mathbb{N}$, and $\mu$ has finite relative entropy with respect to $\gamma_{c}$. Hence $\mu\in\mathcal{P}^{a}_{\rho}$ by Noise-Optimal Maps out of the Entropy Domain of a Diagonal Gaussian Reference Measure §connected.

Claim 2. Let $\mathcal{Q}_{c}$ be the set of the $\mu\in\mathcal{P}^{a}_{\rho}$ that have a density with respect to $\gamma_{c}$, as defined in the statement of Tangency of Noise-Optimal Displacements out of a Measure with a Density Relative to a Diagonal Gaussian Measure, and the Noise Map Property, whose data are present here with this $c$ and the reference measure $\rho=\gamma_{c}$. Let $\mu\in\mathcal{D}_{H}$. By Relative Entropy of Probability Measures §relative-entropy, $\mu$ has a density with respect to $\gamma_{c}$, and $\mu\in\mathcal{P}^{a}_{\rho}$ by claim 1; so $\mu\in\mathcal{Q}_{c}$. Thus $\mathcal{D}_{H}\subseteq\mathcal{Q}_{c}$.

By Tangency of Noise-Optimal Displacements out of a Measure with a Density Relative to a Diagonal Gaussian Measure, and the Noise Map Property §map-property, $\mathcal{Q}_{c}$ has the noise map property: for every $\mu\in\mathcal{Q}_{c}$ and every $\nu\in\mathcal{P}^{a}_{\rho}$ the ordered pair $(\mu,\nu)$ is uniquely noise-mapped and $T-\mathrm{id}\in T^{a}_{\mu}$ for a noise-optimal map $T$ from $\mu$ to $\nu$, where $\mathrm{id}$ is the identity map of $X$ and $T^{a}_{\mu}$ the noise tangent space at $\mu$, as The Noise Map Property of a Set of Probability Measures §map-property requires. That condition is imposed on each member $\mu$ of the set separately, so it holds in particular for every $\mu\in\mathcal{D}_{H}\subseteq\mathcal{Q}_{c}$; and $\mathcal{D}_{H}\subseteq\mathcal{P}^{a}_{\rho}$ by claim 1. Hence $\mathcal{D}_{H}$ has the noise map property. $\blacksquare$