Why is it enough to consider simple functions on open intervals to prove the Riemann-Lebesgue lemma?

Question

I was looking at the proof of the Riemann-Lebesgue lemma on wikipedia. I noticed that in their proof they assumed it was enough to show that it held for simple functions of the form: $$\sum_{n=1}^m c_n\chi_{(a_n,b_n)}$$ before applying that simple functions are dense in $L^1$. But can't we have simple functions of the form $\sum_{n=1}^m c_n\chi_{E_n}$, where $E_n$ is a Borel set not necessarily an interval? How do we know the assertion holds for these simple functions?

Answer 1

You may need to look the fundamental properties of Lebesgue measure, in particular , the Lebesgue measure is regular. This means that for any Borel set $A$, their exists a sequence of open sets $(\mathcal U_n)_{n\in \mathbb N}$ such that $$\forall n\in \mathbb N, \quad\quad A \subset \mathcal U_n \text{ and } \lambda(\mathcal U_n\setminus A) \leq \frac{1}{n}.$$ This allows you to approximate in $L^1$ any simple function by step functions.

To conclude : $\{\text{Step functions}\}$ is dense in $\{\text{simple functions}\}$ which is dense in $L^1$