The Wikipedia proof of Radon-Nikodym first shows for positive measures, then extends to real measures by applying the positive measure Radon-Nikodym theorem to the unique + and - (positive) measures obtained via Hahn-Jordan decomposition. But one crucial assumption of Radon-Nikodym is that the (positive) measures are absolutely continuous w.r.t. $\mu$.
My question is, if $\nu \ll \mu$ how do we know $\nu^+ \ll \mu$? Isn't it possible that $\nu(E) = 0$ but $\nu^+(E) = \nu^-(E) > 0$?