If $X$ is a complete doubling metric space equipped with a complete probability measure $\mu$ such that all Borel sets are $\mu$-measurable, then $C_c(X)$ — the continuous functions with compact support — are dense in $L_1(\mu)$.
Question: What are the weakest conditions under which $C_c(X)$ is dense in $L_1(\mu)$ for non-doubling (i.e., infinite doubling dimensional) metric spaces?
Best Answer
Like in Nate's comment, you need locally compactness. Assuming that, if $\mu$ is regular, then $C_c(X)$ is dense in $L^1(\mu)$. For the proof: For a compact subset $K$ there exists a sequence $f_n$ in $C_c(X)$ with $f_n\ge 1_K$ and $\int f_n\,d\mu\to \mu(K)$. Therefore, $1_K$ lies in the closure of $C_c(X)$ in $L^1$. By another approximation, $1_A$ lies in this closure for every measurable $A$. This suffices.