Coupling

Remarks

Example: the product measure $\mu \otimes \nu$ is a coupling, the independent coupling. Indeed $(\mu \otimes \nu)(A \times F) = \mu(A) \nu(F) = \mu(A)$, and in the same way for the second marginal. Example: for $0 \le p \le q \le 1$, let $U$ be uniform on $[0, 1]$, $X = \mathbf{1}_{\set{U < p}}$ and $Y = \mathbf{1}_{\set{U < q}}$. Then $X$ and $Y$ are Bernoulli with parameters $p$ and $q$, and $X \le Y$ everywhere. Non-example: on $\set{0, 1}^2$, the point mass $\delta_{(0, 0)}$ is not a coupling of two Bernoulli laws with parameter $\frac{1}{2}$, because its first marginal is $\delta_0$. If $E = F$ and the diagonal is measurable, every coupling satisfies the coupling inequality $d_{TV}(\mu, \nu) \le P(X \ne Y)$, where $d_{TV}$ is the total variation distance.