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?

infinite dimensionalBanach space, an open ball is not precompact, and the support of a nonzero continuous function contains some open ball, so it is not compact. I'm not sure what separable had to do with anything. $\endgroup$9more comments