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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.

Filippo Giovagnini
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  • Here (https://math.stackexchange.com/questions/536362/countably-generated-sigma-algebra-implies-separability-of-lp-spaces?noredirect=1&lq=1) is an answer with the additional assumption that the measure space is $\sigma$-finite. – Davide Giraudo May 01 '20 at 21:59
  • @DavideGiraudo Thank you so much, I will read it! – Filippo Giovagnini May 02 '20 at 07:29

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