Are continuous functions dense in $L^1$?

Question

It is a well known fact that the continuous compactly supported functions are dense in $L^1(\mathbb R)$.

An immediate counterexample to this fact for a non locally compact space is $\mathbb R \setminus \mathbb Q$ with the restricted Lebesgue measure.

But are continuous functions always dense in $L^1(X)$, for any metric space $X$ with a measure on the Borel $\sigma$-algebra? Are there any counterexamples?

Answer 1

Try $X = \mathbb R$ with the usual topology, and counting measure on the rationals. The only continuous function in $L^1$ is $0$.