Exercise 3.6.2 of Durrett’s *Probability: Theory and Examples*, 4th edition is

Show that $\|\mu-\nu\| \le 2\delta$ if and only if there are random variables $X$ and $Y$ with distributions $\mu$ and $\nu$ so that $P(X \ne Y) \le \delta$.

The first inequality should read $\|\mu-\nu\| \le \delta$.