C_c(X) is dense in L^p(mu) for a Radon measure

It suffices by [L1] to approximate ${\mathbf{1}}_E$ when $\mu (E)<\mathrm{\infty }$. [L1, choose] Given $\eta >0$, outer regularity gives open $U\supseteq E$ with $\mu (U\setminus E)<{\eta }^p\mathrm{/}2$, and inner regularity of $U$ gives compact $K\subseteq U$ with $\mu (U\setminus K)<{\eta }^p\mathrm{/}2$.

Choose $f\in C_c(X)$ with ${\mathbf{1}}_K\le f\le {\mathbf{1}}_U$ by [L2]. [step 1.1, L2] Then $\mathrm{\mid }f-{\mathbf{1}}_E\mathrm{\mid }\le {\mathbf{1}}_{U\setminus K}+{\mathbf{1}}_{U\setminus E}\mathrm{.}$ Since both indicators take only the values zero and one and their union is $U\setminus (K\cap E)$, while $0\le f,{\mathbf{1}}_E\le 1$, the error is bounded by the indicator of that union. Hence $\mathrm{\parallel }f-{\mathbf{1}}_E{\mathrm{\parallel }}_p^p\le \mu (U\setminus K)+\mu (U\setminus E)<{\eta }^p\mathrm{.}$ Thus $\mathrm{\parallel }f-{\mathbf{1}}_E{\mathrm{\parallel }}_p<\eta$.

Approximate the finitely many indicator terms of a simple function separately and sum the resulting $C_c$ functions. Then use [L1] and the triangle inequality.