Is $C_c$ dense in $L_p$ for $0<p<1$?

Question

Let $C_c$ be the set of compactly supported functions on $\mathbb{R}$ that are infinitely differentiable. Let $S$ be the set of Schwartz functions. It is well known that $C_c$ (hence $S$) is dense in $L_p$, $p\in[1,\infty$). What can be said about the case $0<p<1$? Is $C_c$ or $S$ also dense in these spaces?

Answer 1

Yes, it is. First show that the set of simple functions $f=\sum a_n \chi_{A_n}$ is dense, where the sum is finite and each $A_n$ is of finite measure.

Then use regularity of Lebesgue measure to find, for $A$ measurable of finite measure, compact $K$ and open $U$ with $K\subset A\subset U$ and $\mu(U\setminus K)<\epsilon$.

Using what is called the $C^\infty$ Urysohn-Lemma, find $f \in C_c^\infty(U)$ with $f=1$ on $K$ and $0\leq f \leq 1$.

Verify that $\Vert f -\chi_A\Vert_p$ is small.

Put the pieces together to conclude the proof.