The Noise Map Property of a Set of Probability Measures

In the setting of Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation, let $\mathcal{P}^{a}_{\rho}$ be the set of The Measures Noise-Connected to the Reference Measure §space. For $\mu,\nu\in\mathcal{P}^{a}_{\rho}$ 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, so that the noise Wasserstein distance $W_{a}(\mu,\nu)$ and the noise-optimal couplings of $\mu$ and $\nu$ are defined. Noise-optimal maps $T$ from $\mu$ to $\nu$, their displacements $T-\mathrm{id}\in L^{2}(\mu;X^{a})$ and uniquely noise-mapped pairs are those of Noise-Optimal Maps and Uniquely Noise-Mapped Pairs, $\mathrm{id}$ is the identity map of $X$, and $T^{a}_{\mu}\subseteq L^{2}(\mu;X^{a})$ is the noise tangent space at $\mu$.

1. (Noise map property)¶ A set $Q\subseteq\mathcal{P}^{a}_{\rho}$ has the noise map property if for all $\mu\in Q$ and $\nu\in\mathcal{P}^{a}_{\rho}$ the ordered pair $(\mu,\nu)$ is uniquely noise-mapped and, $T$ denoting a noise-optimal map from $\mu$ to $\nu$,

$$T-\mathrm{id}\in T^{a}_{\mu}.$$

This condition does not depend on the choice of $T$. Indeed, let $T$ and $T'$ be noise-optimal maps from $\mu$ to $\nu$, the pair being uniquely noise-mapped. The couplings $(\mathrm{id},T)_{\#}\mu$ and $(\mathrm{id},T')_{\#}\mu$ are both noise-optimal by Noise-Optimal Maps and Uniquely Noise-Mapped Pairs §map, and every noise-optimal coupling of $\mu$ and $\nu$ equals $(\mathrm{id},T_{0})_{\#}\mu$ for the map $T_{0}$ of Noise-Optimal Maps and Uniquely Noise-Mapped Pairs §uniquely-mapped; hence $(\mathrm{id},T)_{\#}\mu=(\mathrm{id},T_{0})_{\#}\mu=(\mathrm{id},T')_{\#}\mu$. Writing $z=(x,y)$ with $x=\pi_{1}(z)$ and $y=\pi_{2}(z)$, the function $z\mapsto\min\{1,|y-T(x)|\}$ on $X\times X$ is $g\circ(T\circ\pi_{1},\pi_{2})$ with $g:X\times X\to\mathbb{R}$, $g(u,v)=\min\{1,|v-u|\}$. Here $\pi_{1},\pi_{2}$ are Borel by Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §product-sigma, $T\circ\pi_{1}$ is Borel by claim 4 of Borel Measurability and Bounded Integration on a Metric Space, and the pair $(T\circ\pi_{1},\pi_{2})$ is Borel by Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §pairing; and $g$ is continuous, indeed Lipschitz with constant $2$, since $|\min\{1,s\}-\min\{1,t\}|\le|s-t|$ for real $s,t$ and, by the triangle inequality of the norm, $\bigl||v-u|-|v'-u'|\bigr|\le|v-v'|+|u-u'|$, so that $g$ is Borel by claim 3 of Borel Measurability and Bounded Integration on a Metric Space. Hence the function is Borel by claim 4 of Borel Measurability and Bounded Integration on a Metric Space; likewise $x\mapsto\min\{1,|T'(x)-T(x)|\}$, the composite of this function with the Borel map $(\mathrm{id},T')$ (Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §pairing), is Borel by the same claim. The function on $X\times X$ is bounded, hence integrable against both measures by claim 6 of Borel Measurability and Bounded Integration on a Metric Space. By the change-of-variables formula its integral against $(\mathrm{id},T)_{\#}\mu$ is the integral of the zero function, which is $0$, and its integral against $(\mathrm{id},T')_{\#}\mu$ is $\int_{X}\min\{1,|T'(x)-T(x)|\}\,\mu(dx)$; the two are equal. By The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §vanishing the nonnegative Borel function $x\mapsto\min\{1,|T'(x)-T(x)|\}$ vanishes for $\mu$-almost every $x$, that is, $T'(x)=T(x)$ for $\mu$-almost every $x$, so the classes of $T-\mathrm{id}$ and of $T'-\mathrm{id}$ in $L^{2}(\mu;X^{a})$ coincide by The Space of Square-Integrable Maps from a Measure Space into a Hilbert Space with an Orthonormal Basis §classes.