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!