Understanding the proof that $C_{c}(E)$ is dense in $L^p(E)$ (Royden and Fitzpatrick "Real Analysis" 4th edition page 153)

Question

Here is the theorem and a hint for the proof before it:

But I still do not know a thorough proof for it, proposition 9 proved that simple functions are dense in $L^p(E)$, but then how to proceed that continuous functions are dense in $L^p(E)$ or $L^1([0,1] , m)$ where $m$ is the Lebesgue measure, that is all what I need, could anyone help me in this please?

Answer 1

In Rudin's Real and Complex he proves that $C_c(\mathbf{R})$ is dense in the space of simple functions. Density is a transitive relation so we get $C_c(\mathbf{R})$ is dense in $L^p(\mathbf{R})$