Fibrewise Cyclical Monotonicity of a Noise-Optimal Coupling: Given the Tail, the Heads are Cyclically Monotone

In the setting of Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation, with the coordinates $x_{k}$ and the maps $Q_{n}$ of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §coordinates, and with $x=\pi_{1}(z)$ and $y=\pi_{2}(z)$ for $z\in X\times X$ as in Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §background, let $\mu,\nu\in\mathcal{P}(X)$ be such that the ordered pair $(\mu,\nu)$ is noise-connected, and let $\pi$ be a noise-optimal coupling of $\mu$ and $\nu$. Let $n\in\mathbb{N}$. The map $t_{n}=Q_{n}\circ\pi_{1}:X\times X\to X$ is Borel by Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §continuity, 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; let $\mu_{n}=(t_{n})_{\#}\pi$, which is $(Q_{n})_{\#}\mu$ since $(\pi_{1})_{\#}\pi=\mu$, and let $\kappa$ be a conditional kernel of $\pi$ given $t_{n}$, with $\kappa_{w}=\kappa(w,\cdot)$.

Let $\mathbb{R}^{n+n}$ be as in Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pairs, and let $h_{n}:X\times X\to\mathbb{R}^{n+n}$ be the map whose first $n$ components are $x_{1},\dots,x_{n}$ and whose last $n$ components are $a_{1}^{-1}y_{1},\dots,a_{n}^{-1}y_{n}$. For $k\in\mathbb{N}$ the functions $x\mapsto x_{k}$ and $x\mapsto a_{k}^{-1}x_{k}$ on $X$ are Lipschitz, with constants $1$ and $a_{k}^{-1}$, by Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §continuity, hence Borel by claim 3 of Borel Measurability and Bounded Integration on a Metric Space; so each component of $h_{n}$, the composite of one of them with the Borel map $\pi_{1}$ or $\pi_{2}$ (Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §product-sigma), is Borel by claim 4 of Borel Measurability and Bounded Integration on a Metric Space, and $h_{n}$ is Borel by claims 2 and 5 of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets. For $w\in X$ let $\theta_{w}=(h_{n})_{\#}\kappa_{w}$, a Borel probability measure on $(\mathbb{R}^{n+n},d_{E})$, and let $\operatorname{supp}\theta_{w}$ be its support.

1. (Fibrewise cyclical monotonicity)¶ There is a set $W_{1}\in\mathcal{B}(X)$ with $\mu_{n}(W_{1})=1$ such that, for every $w\in W_{1}$, the set $\operatorname{supp}\theta_{w}$ is cyclically monotone.