Let $\mu$ and $\nu$ be two distributions on the state space, $S$, of a Markov chain.
The total variation distance between $\mu$ and $\nu$ is defined as $$||\mu - \nu|| = \underset{A \subseteq S}{\sup} \left( \mu(A) - \nu(A) \right).$$
Does there necessarily exist some $A' \subseteq S$ such that $$\mu(A') - \nu(A') = ||\mu - \nu||?$$ This seems non-obvious to me, because, in general, the supremum of a set may or may not be an element of the set.