Head Maps of a Noise-Optimal Coupling: the Head of the Target is a Function of the Source

In the setting of Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation, with the maps $p_{n}$ of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §coordinates, which are Borel by Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §continuity, let $c$ be a variance sequence with diagonal Gaussian measure $\gamma_{c}$, and let $\mu,\nu$ belong to the set $\mathcal{P}^{a}_{\rho}$ of The Measures Noise-Connected to the Reference Measure §space, so that the ordered pair $(\mu,\nu)$ is noise-connected by The Noise Wasserstein Distance is a Metric on the Measures Noise-Connected to the Reference Measure: Existence of Noise-Optimal Couplings, Comparison with the Quadratic Wasserstein Distance and Lower Semicontinuity §connected. Suppose that $\mu$ has a density with respect to $\gamma_{c}$, and let $\pi$ be a noise-optimal coupling of $\mu$ and $\nu$. Let $n\in\mathbb{N}$, write $\mathrm{id}$ for the identity map of $X$, and let $X\times\mathbb{R}^{n}$ carry the product $\sigma$-algebra $\mathcal{B}(X)\otimes\mathcal{B}(\mathbb{R}^{n})$. The components $\pi_{1}$ and $p_{n}\circ\pi_{2}$ of the map $(\pi_{1},p_{n}\circ\pi_{2}):X\times X\to X\times\mathbb{R}^{n}$ are Borel by Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §product-sigma and claim 4 of Borel Measurability and Bounded Integration on a Metric Space. By Pairings into a Product, the Graph of a Measurable Map, and Couplings Concentrated on a Graph §pairing, applied with $(X,\mathcal{B}(X))$ and $\mathbb{R}^{n}$ in place of $(Y,\mathcal{Y})$ and $Z$, this map and the map $(\mathrm{id},S)$ for every Borel $S:X\to\mathbb{R}^{n}$ are measurable into $X\times\mathbb{R}^{n}$; image measures are those of claim 1 of Image Measures, Measures with Densities, and Change of Variables.

1. (The head of the target is a function of the source)¶ There is a Borel map $T_{n}:X\to\mathbb{R}^{n}$ with

$$(\pi_{1},p_{n}\circ\pi_{2})_{\#}\pi=(\mathrm{id},T_{n})_{\#}\mu .$$

2. (Almost sure form)¶ Let $T_{n}$ be as in claim 1. The set $G_{n}=\{z\in X\times X:\ p_{n}(y)=T_{n}(x)\}$, where $x=\pi_{1}(z)$ and $y=\pi_{2}(z)$, is the preimage under $(\pi_{1},p_{n}\circ\pi_{2})$ of the graph of $T_{n}$, which belongs to $\mathcal{B}(X)\otimes\mathcal{B}(\mathbb{R}^{n})$ by Pairings into a Product, the Graph of a Measurable Map, and Couplings Concentrated on a Graph §graph-measurable, $\mathbb{R}^{n}$ being separable by Euclidean Space is a Separable Metric Space §separable; so $G_{n}\in\mathcal{B}(X\times X)$, and $\pi(G_{n})=1$.