Exercise 3.6.2 of Durrett’s Probability: Theory and Examples, 4th edition is
Show that ∥μ − ν∥ ≤ 2δ if and only if there are random variables X and Y with distributions μ and ν so that P(X≠Y) ≤ δ.
The first inequality should read ∥μ − ν∥ ≤ δ.
Errata: Durrett, Exercise 3.6.2
May. 16, 2015