Couplings of Finite Noise Cost and Their Noise Cost

In the setting of Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation, let $\mu,\nu\in\mathcal{P}(X)$, with $\Pi(\mu,\nu)$ the set of their couplings. By The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §pairs the set $D_{a}$ belongs to $\mathcal{B}(X\times X)$ and the function $c_{a}:X\times X\to\mathbb{R}$ is Borel and nonnegative, so that for every $\pi\in\Pi(\mu,\nu)$ the number $\pi(D_{a})$ is defined and the integral $\int_{X\times X}c_{a}\,d\pi$ of a nonnegative Borel function, as in Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §measures, is defined in $[0,\infty]$.

1. (Finite noise cost)¶ A coupling $\pi\in\Pi(\mu,\nu)$ has finite noise cost if $\pi(D_{a})=1$ and $\int_{X\times X}c_{a}\,d\pi<\infty$.

2. (Noise cost)¶ The noise cost of a coupling $\pi\in\Pi(\mu,\nu)$ of finite noise cost is the nonnegative real number

$$I^{a}(\pi)=\int_{X\times X}c_{a}\,d\pi .$$

3. (Couplings of finite noise cost)¶ $\Pi^{a}(\mu,\nu)$ denotes the set of the couplings $\pi\in\Pi(\mu,\nu)$ of finite noise cost.

4. (Noise-connected measures)¶ The ordered pair $(\mu,\nu)$ is noise-connected if $\Pi^{a}(\mu,\nu)$ is nonempty.