Dense subspaces of $L^p$ for $1\leq p<\infty$

Let \(1\leq p<\infty\) and \(n\geq1\). Each of the following subspaces is dense in \(L^p(\RR^n)\) [@Fol13]:

1. simple functions \(\sum_{j=1}^k a_j\chi_{E_j}\) with \(a_j\in\CC\) and \(E_j\) Lebesgue measurable of finite measure;

2. step functions \(\sum_{j=1}^k a_j\chi_{R_j}\) with \(a_j\in\CC\) and \(R_j\) closed rectangles;

3. bounded measurable functions with bounded support;

4. \(C_c(\RR^n)\), the continuous functions with compact support;

5. \(C_c^\infty(\RR^n)\), the smooth functions with compact support.

For a bounded interval \([a,b]\), \(C([a,b])\) and the functions in \(C_c^\infty((a,b))\) extended by \(0\) are dense in \(L^p([a,b])\).