Given two probability measures $\mu, \nu$ on a probability space $X$, their total variation is defined by $$ ||\mu - \nu|| := 2 \sup_{A \subset X}|\mu(A)-\nu(A)|.$$ I would like to know: If $||\mu - \nu|| = 2$ does it imply that the measures $\mu$ and $\nu$ are not absolutely continuous with respect to one another?
I tried to construct a set $A$ that is a null set for $\mu$ but not for $\nu$ however I can't figure out how to do this. Any help is appreciated!