Let $(\Omega, \Sigma)$ be some measurable space with measures $\nu$ and $\mu$. So long as $\nu \ll \mu$ it makes sense to think of $L^2(\mu) \cap L^2(\nu)$ as a subspace of $L^2(\mu)$ since they'll have the same underlying set. Is there a general criteria for $L^2(\nu)$ to actually be dense in $L^2(\mu)$? What if we specialise to the case $\Omega = \mathbb{R}^n$ and $\mu$ the Lebesgue measure?
Edit: To provide some motivation, I'm wondering in which situations it makes sense to define $-\Delta + V$ as an unbounded operator on a Hilbert space where $V$ is a multiplication operator. $-\Delta$ naturally acts on $H^1$ as an unbounded operator and I'm wondering if there's a general criterion on $V$ so that $L^2( V(x) dx )$ is dense in $L^2$.