On page 99 of Folland's Real Analysis, he states that
Let $\nu$ be a regular signed or complex Borel measure on $\mathbb{R}^{n}$, if $d\nu = d\lambda +fdm$ is a Lebesgue-Radon-Nikodym representation of $\nu$, where $m$ is the Lebesgue measure ... It is easily verified that $d|\nu| = d|\lambda|+|f|dm$.
I think for any signed or complex sigma finite measure $\nu$ and a sigma finite positive measure $\mu$, if $d\nu = d\lambda+fd\mu$ is the Lebesgue-Radon-Nikodym representation of $\nu$, then we also have $d|\nu| = d|\lambda| + |f|d\mu$
Proof sketch: We have $\lambda \ll |\lambda|$ and $fd\mu \ll |fd\mu| = |f|d\mu$, thus if we set $dn = d|\lambda| + |fd\mu|$, we have $g,h$ such that $\lambda =g \,dn,fd\mu = h \,dn$ by radon-nikodym. As $\lambda \bot f \, d\mu$, we also have $|\lambda| \bot |f|d\mu$. Let $|\lambda|$ null of $F$ and $|f|d\mu$ null on $E$ ,where $E\cap F = \emptyset$ and $E \cup F = X$. we can write $\lambda = g \chi_{E} \,dn$, $fd\mu = h \chi_{F} \, dn$, and therefore as $d\nu = g\chi_{E}+h\chi_{F}\, dn$$$d|\nu| = |g\chi_{E} +h\chi_{F}| dn = |g\chi_{E}|dn+ |h\chi_{F}|dn = |\lambda| + |f|d\mu$$
Is this statement true, and does the proof work?