Variation Signed Measure Inequality

Question

Given $\nu_1,\nu_2$ finite signed measures, is there a way to prove $|\nu_1 + \nu_2| \le |\nu_1| + |\nu_2|$ without resorting to the fact that for a general signed measure $\nu$, $$|\nu|(E) = \sup \sum_{i=1}^{n}|\nu(E_i \cap E)|,$$ where the $E_i$ are disjoint, measurable?

Answer 1

One way you may want to try is using $\nu = \nu^+-\nu^-$ and $\nu=\nu^++\nu^-$, the Jordan decomposition. This could simplify the problem a little bit. But as @Lolman suggest, you can not actually avoid the definition of total variation, as the above identity is proved by the definition of total variation.