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

In the setting of Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation, let $c$ be a variance sequence and suppose that the reference measure is $\rho=\gamma_{c}$, the diagonal Gaussian measure on $X$ with variances $c$, which is admissible since $\gamma_{c}\in\mathcal{P}_{2}(X)$ by Moments of a Diagonal Gaussian Measure on a Hilbert Space: Coordinate Covariances, Finite Second Moment, and Exponential Moments of the Squared Norm §moment. Let $\kappa$ be a positive real number with $c_{k}\le\kappa\,a_{k}$ for every $k\in\mathbb{N}$, where $a$ is the sequence of noise weights. Let $\mathcal{P}^{a}_{\rho}$ be the set of The Measures Noise-Connected to the Reference Measure §space, and let $\mathcal{D}_{H}$ be the set of the $\mu\in\mathcal{P}(X)$ that have finite relative entropy with respect to $\gamma_{c}$.

1. (Inclusion)¶ $\mathcal{D}_{H}\subseteq\mathcal{P}^{a}_{\rho}$.

2. (The noise map property)¶ $\mathcal{D}_{H}$, which is a subset of $\mathcal{P}^{a}_{\rho}$ by claim 1, has the noise map property.