Let $(X,\Omega)$ be a measurable space and let $\mu, \nu$ be two $\sigma$-finite measures on $(X,\Omega)$. Suppose $\nu \ll \mu$ and let $\phi$ be the Radon-Nikodym derivative of $\nu$ with respect to $\mu$ $(\phi = d\mu/d\nu)$. Define $V:L^2(\nu)\rightarrow L^2(\mu)$ by $Vf=\sqrt \phi f$. Show that $V$ is a well-defined linear isometry and $V$ is an isomorphism if and only if $\mu \ll \nu$ (that is, if and only if the measures are mutually absolutely continuous.)
This is exercise 5.8 in chapter 1 of Conway's A Course in Functional Analysis, second edition. It can be found on page $23$.