I am going through Bogachev's book on Measure Theory, and having read the definition of variation of measures I am trying to grasp what it is doing. I have the following doubt.
Let $\nu$ be a signed measure and define its variation as $|\nu|:=\nu^++ \nu^-$ where $(\nu^+,\nu^-)$ is the Hahn-Jordan decomposition of $\nu$ (e.g. p. 176 in Bogachev's book). If $\nu$ is given by $\nu=m_1-m_2$, where $m_1$ and $m_2$ are two positive finite measures, is it true that $|\nu|=m_1 + m_2$?
I could not prove the equality, however since $m_1\geq\nu^+$ and $m_2\geq\nu^-$ (e.g. Components of signed measure are greater than or equal to positive and negative variation of the signed measure.) it seems that $|\nu|\leq m_1 + m_2$ holds. Is the other inequality true?