A coupling of two probability measures $P,\tilde P$ on a Borel space $X$ is any probability measure on $X^2$ whose one-dimensional marginals are $P$ and $\tilde P$. In particular, for any such coupling $\Bbb P$ we have $$ 2\cdot \Bbb P(X^2\setminus\Delta_X)\geq\|P - \tilde P\| \tag{1} $$ where $\Delta_X = \{(x,x):x\in X\}$ is the diagonal, and $\|\cdot\|$ denotes the total variation norm. At the same time, there always exists a maximal coupling $\Bbb P^*$ such that the equality holds in $(1)$.

Extension of these results to the case when $P$ and $\tilde P$ are non-probability measures, or at least sub-probabilities does not seem to be complicated, however I wonder whether it has been already addressed somewhere. So far a search on google did not help.

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