Are the continuous functions on $G$ dense in $L^{1}(G)$?

Question

If $G$ is a locally compact group, is the set $C_{c}(G)$ of all continuous functions on $G$ with compact support dense in $L^{1}(G)$?

Answer 1

Yes. A more general result is the following:

Proposition. Let $X$ be a locally compact Hausdorff space, and let $\mu$ be a Radon measure on $X$. Then for any $p \in (0, \infty)$, $C_c(X)$ is a dense subset of $L^p(X, \mu)$.

See Proposition 3 in Terry Tao's blog post for a proof (this post also appears as a chapter in Tao's book An Epsilon of Room). Another reference is Proposition 7.9 in Folland's Real Analysis.