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

Thanks!