Problem involving decomposition of measures Let $\mu$ be a signed measure. We wish to prove that $$\left| \int{f} \> d\mu \right| \leq \int{|f|} \> d|\mu|.$$
(We are given the following defintion: $\int{f} \> d\mu = \int{f} \> d\mu^{+} - \int{f} \> d\mu^{-}$.)
For the proof, I want to just write $$\left| \int{f} \> d\mu \right| = \left| \int{f} \> d\mu^{+} - \int{f} \> d\mu^{-} \right| \leq \left| \int{f} \> d\mu^{+} \right| + \left| \int{f} \> d\mu^{-} \right| \leq \int{|f|} \> d\mu^{+} + \int{|f|} \> d\mu^{-} = \int{|f|} \> d|\mu|.$$
But, I feel as though I am missing justification for the last (and, obviously, most important) equality. That is, I'm sure if that follows from the definition provided to me. Any help would be great! 

The last equality rests on the fact that $$|\mu|=\mu^++\mu^-,\tag{1} $$ where $\mu= \mu^+-\mu^-$ (Jordan-Hahn decomposition), where $\mu^+$ and $\mu^-$ are positive and muutually singular. 
Equality (1) is either the definition of $|\mu|$, or it canbe deduced from the definition of $|\mu|$ with partitions using the supports of $\mu^+$ and $\mu^-$.
