Reading "Functional Analysis, Sobolev Spaces and Partial Differential Equations" by Haim Brezis I found a proof of separability of $L^p$ for $p \in [1,+\infty[$ when the measure space is $(\mathbb{R}^N, \mathbb{B}(\mathbb{R}^N), \mathcal{L})$, but the autor says it is true in general when $(\Omega, \mathcal{E})$ is a general separable measure space. How can we prove it?
We say that $(\Omega, \mathcal{E})$ is a separable measure space if there exists a countable family of the $\sigma$-algbera $\mathcal{W} \subset \mathcal{E}$ such that the $\sigma$-algebra generate by $\mathcal{W}$ is exactly $\mathcal{E}$.
Thanks in advance.