Is there a sequence of $\{f_n\} \subset C^{\infty}_{c}(\mathbb{R}^n)$ which converge to a function $f$ in $L^2(\mu)$?

Question

I am trying to figure out if there is a sequence of $\{f_n\} \subset C^{\infty}_{c}(\mathbb{R}^n)$ which converges to a function $f$ in $L^2({\mu})$, where $\mu$ is a probability measure, i.e., $\mu(dx) = \rho(x)dx$ on $\mathbb{R}^n$ and $\rho(x)$ is a density function.

I know if $\mu$ is just Lebesgue measure($dx$), there is a convergent sequence of $\{f_n\} \in C^{\infty}_{c}(\mathbb{R}^n)$ which converges to a function $f$ in $L^2(\mathbb{R}^n)$, but when the measure is a probability measure, it's a bit different. I'd appreciate it if you'd give me any help. Thank you.

Answer 1

For any Borel measure $\mu$ on $\mathbb R^{n}$ which is finite on compact sets $C_c^{\infty}( \mathbb R^{n})$ is dense in $L^{p}(\mathbb R^{n})$ for any $1\leq p<\infty$. This is proved in Rudin's RCA.