Complex Measures: Decomposition Given a complex measure: $\mu:\Sigma\to\mathbb{C}$.



Consider its decomposition into positive measures:
$$\mu=\Re_+\mu-\Re_-\mu+i\Im_+\mu-i\Im_-\mu=:\sum_{\alpha=0\ldots3}i^\alpha\mu_\alpha$$


  Does it split into disjoint regions: $\mu_\alpha(E)=|\mu|(E\cap A_\alpha)\quad(A_\alpha\cap A_\beta=\varnothing)$
Sure. Set $A$ to be the set on which $\mu$ is positive and $B$ the set on which its negative.
Consider the Radon-Nikodym derivatives:
$$\mu_\alpha(E)=\int_Eu_\alpha\mathrm{d}|\mu|$$
They give rise to the constraints:
$$u_\alpha\geq0:\quad|u|=1\implies u_\alpha=\chi_{A_\alpha}$$


  So the above can only hold for: $\mathrm{im}u\subseteq\{1,i,-1,-i\}$


As an example when this fails consider:
$$\Omega:=[0,1]:\quad\mu(A):=\int_Ae^{it}\mathrm{d}t$$
(This way it becomes clear since it takes other values, too.)