I was looking for a simple application of the Stone-Weierstrass theorem.
First I thought that if $X$ is any compact measure space then Stone-Weierstrass implies that $C_c(X)$ is dense in $L^p$.
But I have to assume that $X$ is compact otherwise I don't have $1$ in $C_c(X)$. That of course makes it a boring example since then $C_c(X) = C(X)$. Can someone show me a slightly more interesting but still simple example? Thank you.