Suppose $\nu$ is a regular signed or complex Borel measure on $\mathbb R^n$, *m* is the Lebesgue measure on the class of Borel sets $\mathcal B\_{\mathbb R^n}$ and the Lebesgue-Radon-Nikodym decomposition of $\nu$ is $d\nu=d\lambda+fdm$ where *f* is an extended *m*-integrable function when $\nu$ is a signed measure or $\in L^1(m)$ when $\nu$ is a complex measure and $\lambda\bot m$. Prove that $d|\nu|=d|\lambda|+|f|dm$ where the notation $|\bullet|$ represents total variation. 


PS: A Borel measure $\nu$ on $\mathbb R^n$ is regular if $\nu(K)<\infty$ for every compact *K*. From this definition we have $\nu(E)=\inf \{\nu(U)|U{\;\rm{ open},}U\supseteq E\}$. A signed or complex Borel measure on $\mathbb R^n$ is regular if $|\nu|$ is regular.


Thanks!



A question about regular signed or complex Borel measure under LRN decomposition
If $\nu\bot\lambda$, then $|\nu+\lambda|=|\nu|+|\lambda|$


