The Measures Noise-Connected to the Reference Measure

In the setting of Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation, noise-connectedness is that of Couplings of Finite Noise Cost and Their Noise Cost, and $\rho$ is the reference measure of Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §reference.

1. (Measures noise-connected to the reference measure)¶ $\mathcal{P}^{a}_{\rho}$ denotes the set of the $\mu\in\mathcal{P}(X)$ such that the ordered pair $(\mu,\rho)$ is noise-connected; its members are said to be noise-connected to $\rho$.