[Math] Proof of separability of $L^p$ spaces

Question

The following is a proof in Brezis book. It shows the separability of $L^{p}$ spaces:

I have a few questions regarding the proof:

1. It says 'it is easy to construct a function $f_{2} \in \varepsilon…$" and it also says " it suffices to split $R$ into small cubes…'. Would it work to choose $f_{2}$ in the following way:

   Assume we split $R$ as suggested. Let $R_{i}$ denote each small cube of $R$, consider $f_{2_{i}} := C_{i}\chi_{R_{i}}$ where $C_{i}$ is a constant chosen from $[0, \delta – (\text{sup} f|_{R_{i}} – \text{inf} f|_{R_{i}})$, then let $f_{2}(x) := \sum_{i}f_{2_{i}}(x)$. It would then follow that $\Vert f_{1} – f_{2} \Vert_{\infty} < \epsilon$. Is this fine?

2. Can anyone see how the inequality $\Vert f_{1} -f_{2} \Vert_{p} \leq \Vert f_{1}-f_{2} \Vert_{\infty}|R|^{\frac{1}{p}}$ is obtained?

3. Where exactly is the separability of $\Omega = \mathbb{R}^{N}$ used?

Note that $\chi$ denotes the characteristic function.

Thanks a lot for any assistance. Let me know if something is unclear.

Answer 1

1. As you mentioned in a comment, you should choose $C_i \in \Big[ \inf f_1 \restriction R_i, \sup f_1 \restriction R_i \Big]$. This way you get $\big\|f_1 -f_2\big\|_\infty < \delta$. An adequate choice of $\delta > 0$ (that is $\delta |R|^{\frac{1}{p}} < \varepsilon$) would then give you $\big\| f_1 - f_2\big\|_p \leq \varepsilon$.
2. $$ \big\| f_1 - f_2\big\|_p = \left(\int_R |f_1 - f_2|^p \right)^{\frac{1}{p}} \\ \leq \left( \int_R \big\| f_1 - f_2\big\|_\infty^p \right)^{\frac{1}{p}} \\ \left(|R|\cdot \big\| f_1 - f_2\big\|_p \right)^{\frac{1}{p}} \\ |R|^{\frac{1}{p}} \cdot \big\| f_1 - f_2\big\|_\infty.$$
3. A metrizable space is separable if and only if it is second-countable. This means that the euclidean topology on $\mathbb{R}^n$ has a countable basis. This countable basis is explicitely described in the first paragraph of the proof and is put to use in the second paragraph.