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$\mu$ 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 .

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