$C_c({\mathbb{R}}^n)$ is dense in $L^p({\mathbb{R}}^n)$ for $1\le p<\mathrm{\infty }$

By [L1], choose a box-step function [L1, L2, given, choose] $s={\sum }_{j=1}^m{a}_j{\mathbf{1}}_{E_j}$ with $\mathrm{\parallel }f-s{\mathrm{\parallel }}_p<\epsilon \mathrm{/}2$. For each $j$, apply [L2] to choose $K_j\subseteq E_j$ and a bounded open set $O_j\supseteq K_j$ with ${\lambda }_n(E_j\setminus K_j),{\lambda }_n(O_j\setminus K_j)<{\left(\frac{\epsilon }{2^{1+1\mathrm{/}p}m(1+{\sum }_{j=1}^m\mathrm{\mid }{a}_j\mathrm{\mid })}\right)}^p\mathrm{.}$

For each $j$, [L3] gives ${\eta }_j\in C_c({\mathbb{R}}^n)$ with [L3, L4, step 1.1, construct, algebra] $0\le {\eta }_j\le 1$, ${\eta }_j=1$ on $K_j$, and $\mathrm{supp}({\eta }_j)\subseteq O_j$. Put $g:={\sum }_{j=1}^m{a}_j{\eta }_j\in C_c({\mathbb{R}}^n)\mathrm{.}$ Since ${\eta }_j-{\mathbf{1}}_{E_j}$ vanishes off $(E_j\setminus K_j)\cup (O_j\setminus K_j)$ and has absolute value at most $1$ there, $\mathrm{\parallel }g-s{\mathrm{\parallel }}_p\le {\sum }_{j=1}^m\mathrm{\mid }{a}_j\mathrm{\mid }{({\lambda }_n(E_j\setminus K_j)+{\lambda }_n(O_j\setminus K_j))}^{1\mathrm{/}p}<\epsilon \mathrm{/}2$ by [L4].

Therefore [step 1.1, step 2.1, algebra] $\mathrm{\parallel }f-g{\mathrm{\parallel }}_p\le \mathrm{\parallel }f-s{\mathrm{\parallel }}_p+\mathrm{\parallel }s-g{\mathrm{\parallel }}_p<\epsilon \mathrm{.}$ Since $g\in C_c({\mathbb{R}}^n)$, the space $C_c({\mathbb{R}}^n)$ is dense in $L^p({\mathbb{R}}^n)$.