let $P,Q$ be two probability measures on $(\Omega, \mathscr {F})$, and let $\nu$ be a $\sigma$-finite measure on the same event space such that $P \ll v, Q \ll v$. Define $\frac{dP}{dv}=p$, $\frac{dQ}{dv}=q$.
The total variation distance between P and Q is then: $$ V(P,Q) = \sup_{A \in \mathscr{F}}|P(A) - Q(A)|= \sup_A \bigg| \int_A(p-q )d\nu \bigg| $$ I'm confused about the following, we may write: $$ V(P,Q) = \frac{1}{2} \int |p-q|d\nu $$ First, how can we bring the absolute value inside the integral and get rid of the supremum, and second, what region are we integrating over now? This statement is made frequently in the book: Introduction to nonparametric Estimation - Tsybakov