total variation distance: $\max_{A \subseteq \mathcal{A}} \left| P(A)-Q(A) \right| = \frac{1}{2} \sum_{x \in \mathcal{A}} \left| P(x) - Q(x)\right|$?

Question

Could you please suggest a proof or a reference that shows a proof of the following equality (not "passing through the densities" as done in Definition of the total variation distance: $ V(P,Q) = \frac{1}{2} \int |p-q|d\nu$?), where left-hand side and right-hand side are both used as definitions of the total variation distance between two probability measures, $P$ and $Q$? \begin{equation} \max_{A \subseteq \mathcal{A}} \left| P(A)-Q(A) \right| = \frac{1}{2} \sum_{x \in \mathcal{A}} \left| P(x) - Q(x)\right| \end{equation}

Answer 1

I would read ``Markov Chains and Mixing Times'', second edition, by David A. Levin, Yuval Peres with contributions by Elizabeth L. Wilmer, Chapter 4. It is available online.