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?



Continuous functions dense in $L_1$
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.


