Dense subsets in $L^1(\mathbb{R})$

Question

Which of the following are dense subsets in metrical space $L^1(\mathbb{R})$?

• set of smooth functions $C_0^{\infty}(\mathbb{R})$ with compact supports;

• set of above-mentioned functions' derivatives $\{f':f\in C_0^{\infty}(\mathbb{R})\}$;

• set of all simple functions with compact supports.

I'd really appreciate your help with this question. How do I check if a set is dense in this space?