If $\lambda$ is a signed measure and $\lambda\ll\mu$, then $\lambda^{+}\ll\mu$ and $\lambda^{-}\ll\mu$

Question

I was trying to prove the Radon-Nikodym theorem for complex measures, so I tried to decompose the complex measure first into its real and imaginary parts, which are signed measures, and then each of them into its positive and negative variation measure, but then I couldn't prove the absolute continuity with respect to $\mu$ of them. I also tried to prove the absolute continuity of the total variation measure. If someone could give some hints on how to approach this problem I would be very grateful.

Answer 1

By Hahn decomposition theorem we can write $\lambda^{+}(E)=\lambda(E \cap A)$ and $\lambda^{-}(E)=-\lambda(E \cap A^{c})$ for some measurable set $A$. From this the result is obvious.