## A question about regular signed or complex Borel measure

Suppose $\nu$ is a regular signed or complex Borel measure on $\mathbb R^n$. By Lebesgue-Radon-Nikodym theorem, $d\nu=d\lambda+fdm$ where $m$ is the Lebesgue measure on $\mathbb R^n$ and $\lambda\bot m$. How to prove $d|\nu|=d|\lambda|+|f|dm$? I can only prove $d|\nu|\leq d|\lambda|+|f|dm$, but can not establish the converse.
Thanks!