Here (Section "Integration in respect to a complex measure"), they say that:
Another approach is to not develop a theory of integration from scratch, but rather use the already available concept of integral of a real-valued function with respect to a non-negative measure. To that end, it is a quick check that the real and imaginary parts μ1 and μ2 of a complex measure μ are finite-valued signed measures. One can apply the Hahn-Jordan decomposition to these measures to split them as $ \mu_1=\mu_1^+-\mu_1^-$ and $ \mu_2=\mu_2^+-\mu_2^- $ where $μ_1^+, μ_1^-, μ_2^+, μ_2^-$ are finite-valued non-negative measures (unique in some sense).
Where does the minus sign come from? What about $\mu_1^+ +\mu_1^-$?
EDIT Further they write
Then, for a measurable function f which is real-valued for the moment, one can define $ \int_X \! f \, d\mu = \left(\int_X \! f \, d\mu_1^+ - \int_X \! f \, d\mu_1^-\right) + i \left(\int_X \! f \, d\mu_2^+ - \int_X \! f \, d\mu_2^-\right) \tag{1} $ as long as the expression on the right-hand side is defined, that is, all four integrals exist and when adding them up one does not encounter the indeterminate ∞−∞. Given now a complex-valued measurable function, one can integrate its real and imaginary components separately as illustrated above and define, as expected, $ \int_X \! f \, d\mu = \int_X \! \Re(f) \, d\mu + i \int_X \! \Im(f) \, d\mu. \tag{2} $
How does (1) related to (2)? Why doesn't something like $[d]\Re(\mu)$ or show up?