Proof of Noise-Optimal Maps out of the Entropy Domain of a Diagonal Gaussian Reference Measure

Each result cited is universally quantified over the data in its own statement, and is applied below with the data named at each use.

We work in the setting of Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation with reference measure $\rho=\gamma_{c}$, and with $c$, $\kappa$ and $\mu$ as in the statement. By Diagonal Gaussian Measures on a Hilbert Space §measure, $\gamma_{c}$ is a Borel probability measure on $X$, so $\mu$ and $\gamma_{c}$ are probability measures on the measurable space $(X,\mathcal{B}(X))$.

Step 1 (Claim 1). The data $c$ (a variance sequence), $\rho=\gamma_{c}$, $\kappa>0$ with $c_{k}\le\kappa\,a_{k}$ for every $k\in\mathbb{N}$, and $\mu\in\mathcal{P}(X)$ with finite relative entropy with respect to $\gamma_{c}$ satisfy exactly the hypotheses of Talagrand's Inequality in the Noise Norm for a Diagonal Gaussian Reference Measure on a Hilbert Space. Its claim Talagrand's Inequality in the Noise Norm for a Diagonal Gaussian Reference Measure on a Hilbert Space §connected, applied with these data, gives $\mu\in\mathcal{P}^{a}_{\rho}$.

Step 2 (A density of $\mu$). By Relative Entropy of Probability Measures §relative-entropy, read on the measurable space $(X,\mathcal{B}(X))$ with $\mu$ in place of $\nu$ and $\gamma_{c}$ in place of $\gamma$, the hypothesis that $\mu$ has finite relative entropy with respect to $\gamma_{c}$ means in particular that $\mu$ has a density $f$ with respect to $\gamma_{c}$, densities being those of The Radon-Nikodym Theorem for a Finite Measure and a Sigma-Finite Measure, and Uniqueness of Densities for the measure space $(X,\mathcal{B}(X),\gamma_{c})$ and the finite measure $\mu$: $f:X\to\mathbb{R}$ is Borel, $0\le f(x)$ for every $x\in X$, and $\mu(A)=\int_{X}\mathbf{1}_{A}\,f\,d\gamma_{c}$ for every $A\in\mathcal{B}(X)$. This is precisely the hypothesis ``$\mu$ has a density with respect to $\gamma_{c}$'' of Brenier's Theorem in the Noise Norm: Noise-Optimal Couplings out of a Measure with a Density Relative to a Diagonal Gaussian Measure are Induced by a Unique Map, which refers to the same notion of The Radon-Nikodym Theorem for a Finite Measure and a Sigma-Finite Measure, and Uniqueness of Densities on the same measure space.

Step 3 (Claim 2). Let $\nu\in\mathcal{P}^{a}_{\rho}$. In the setting of Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation with $\rho=\gamma_{c}$, the variance sequence $c$ has the diagonal Gaussian measure $\gamma_{c}$; $\mu\in\mathcal{P}^{a}_{\rho}$ by Step 1 and $\nu\in\mathcal{P}^{a}_{\rho}$ by choice; and $\mu$ has a density with respect to $\gamma_{c}$ by Step 2. These are the hypotheses of Brenier's Theorem in the Noise Norm: Noise-Optimal Couplings out of a Measure with a Density Relative to a Diagonal Gaussian Measure are Induced by a Unique Map, and its claim Brenier's Theorem in the Noise Norm: Noise-Optimal Couplings out of a Measure with a Density Relative to a Diagonal Gaussian Measure are Induced by a Unique Map §uniquely-mapped, applied with $c$, $\mu$ and $\nu$, shows that the ordered pair $(\mu,\nu)$ is uniquely noise-mapped in the sense of Noise-Optimal Maps and Uniquely Noise-Mapped Pairs §uniquely-mapped, the notion named in the present statement. Since $\nu\in\mathcal{P}^{a}_{\rho}$ was arbitrary, claim 2 follows.