In Rudin's RCA it says the following:
Suppose $\mu$ is a complex measure on a $\sigma$-algebra $\mathfrak{M}$ in $X$. Then there is a measurable function $h$ such that $|h(x)|=1$ for all $x\in X$, and $\mathrm{d}\mu=h\mathrm{d}|\mu|$.
I don't understand the proof of this theorem. At first it says the follows:
It is trivial that $\mu\ll|\mu|$, hence by Radon-Nikodym's Theorem we have $h\in L^1(|\mu|)$ such that $\mathrm{d}\mu=h\mathrm{d}|\mu|$.
I don't understand why Radon-Nikodym Theorem apply here. Indeed, I do know that in some references (wiki, for example), Radon-Nikodym Theorem is simply referred as that $\lambda\ll\mu$ implies an $h$ such that $\mathrm{d}\lambda=h\mathrm{d}\mu$, however Rudin assert this with respect to the Lebesgue Decomposition (RCA page 121). I wonder if these two assertion is equivalent, i.e., how is Radon-Nikodym Theorem applied here? Thanks in advance.