When is $C_{\mathbb{C}}(\Omega)$ dense in $L_{\mathbb{C}}^{2}(\Omega,\mu)$ with respect to the $L^{2}$-norm? Suppose that $\Omega$ is a topological space and $\mu$ a positive Borel measure on $\Omega$. What assumptions on $\Omega$ and $\mu$ do we need in order to conclude that $C_{\mathbb{C}}(\Omega)$ (= space of continuous functions from $\Omega$ to $\mathbb{C}$) is $L^{2}$-norm dense in $L_{\mathbb{C}}^{2}(\Omega,\mu)$? 
For example, does $\Omega$ need to be second countable, or does $\mu$ need to be regular?
What is the most general setting? Most references I encounter only consider the case where $\mu$ is the Lebesgue measure on some Euclidian space.
 A: (I mention $L^p$, $p\in[1,\infty)$ in my answer because things are the same regarding this)
Usually you would want the continuous functions with compact support; otherwise, you will likely have continuous functions which are not in $L^p$. 
The most general setting I know is the case where $X$ is a locally compact space and $\mu$ is a Radon measure on $X$. In that situation, the density of $C_c(\Omega)$ in $L^p(\Omega,\mu)$ follows rather directly from Lusin's Theorem. 
If you remove regularity of the measure all bets are off (since you lose the relation between the measure and the topology).  For instance on $\mathbb C$ you can take $\mu$ to be the counting measure, and then no continuous function (even with compact support!) is in $L^p(\mathbb C,\mu)$. At the other end of the spectrum, you can take $\mu$ to be the Dirac delta, and now every class in $L^p(\mathbb C,\mu)$ contains a continuous function. 
Similarly, if you get creative with the topology, again things can get awkward quickly. For example now take $\Omega$ with the discrete topology (which is Hausdorff, normal, locally compact, etc., etc.) and $\mu$ any diffuse measure. As any function is continuous, the set of all continuous functions in $L^p(\mu)$ is dense (since it's everything!). But the continuous functions with compact support are the functions with finite support, which will fail to be dense as soon as $\mu$ is not an atomic measure. At the other end of the spectrum, if you take $\Omega$ with the trivial topology, only the constants are continuous, so the continuous functions will fail to be dense in $L^p$ for any measure not concentrated at a point. 
