Density of $C_0(X)$ in $L^2(X)$

Question

Let $X$ be a locally compact space and $m$ be a $\sigma$ finite measure on $X$. Consider $C_0(X)$. Let $f \in C_0(X)$ and $g \in L^2(X)$. Then can I conclude that $fg\in L^2$? Or do we need any more assumption for $m$?

Also is $C_0(X)$ dense in $L^2$?