I'm trying to understanding this proof.
My problem is when he says that $\mu=\frac{d\mu}{d\nu}\nu\quad (1)$.
By Radon-Nikodym, we have $\mu(E)=\int_{E}\frac{d\mu}{d\nu}d\nu$. How could the OP get it to the $(1)$ equation?
Looks like the OP suppose that $\frac{d\mu}{d\nu}$ is constant, but is this true? Is OP correct?
The statement $\mu= f\nu$ for some $\nu$-integrable or positive function $f$ conventionally means that $\mu(E)=\int_E f\textrm{d}\mu$. It's simply notation. Note that in the case where $f$ is constant, this agrees with the usual definition of a scaled measure, i.e. if $\mu=c\nu$, then $\mu(E)=c\nu(E)=\int_E c\textrm{d}\nu$.