Let $X$ be a locally compact Hausdorff space and $\mu$ be a Radon probability measure on $X$. Let $C_0(X)$ be the space of continuous functions on $X$ vanishing at infinity and $N_\mu:=\{f\in C_0(X): \int_X |f|^2 d\mu=0\}$. Now I need to find a familiar space for the closure of $C_0(X)/N_\mu$ in the $L^2$ norm.
Edit: I am looking for the closure instead of the space itself.
I have a feeling that the answer would be $L^2(\mu)$, although I might be wrong. The reason being if we have a similar result like Lusin's theorem, we can approximate every measurable function supported on a finite measure by a compactly supported continuous function, and quotienting by $N_\mu$ only gives the equivalence relation in $L^2(\mu)$.
Note: This question appeared while trying to compute the GNS Hilbert space for an abelian $C^*$ algebras.