$\mu$ and $\nu$ are complex measure, and $|\mu|$ is the total variation, that is,
$$|\mu|(E):=\sup\left\{\sum_{i=1}^\infty|\mu(E_i)|, \{E_i\}_{i=1}^{+\infty}\mbox{ is partition of }E\right\}.$$
Is this always true ?
$$|\mu+\nu|(E)\leqslant|\mu|(E)+|\nu|(E)$$
It seems directly use $|A+B|\leqslant|A|+|B|$to the definition of $|\mu+\nu|$. But...this is too easy, I think I make a mistake or miss something .