This is from Royden text book. I guess here $E \subset P_{\lambda}$, and $\mu(P_{\lambda}) = 0 \implies \mu(E) = 0 \implies v(E) = 0$. The last $\implies$ results from absolute continuity. But, why if $\mu(E)>0$, then $v(E) = 0$? Could you explain this?
Thank you in advance.