If $\mu$ is sigma-finite and $\mathcal{A}$ is countably generated, then $L^p(\mu )$ is separable for $1\le p<\mathrm{\infty }$

Let ${\mathcal{A}}_0$ be the countable algebra from [L2], and let $\mathcal{S}$ [L2, L3, given, algebra] be the set of all finite linear combinations ${\sum }_{j=1}^m{q}_j{\mathbf{1}}_{A_j}$ with $q_j\in \mathbb{Q}(i)$ and $A_j\in {\mathcal{A}}_0$ satisfying $\mu (A_j)<\mathrm{\infty }$ for every $j$. Because both the coefficient set and the finite-measure members of ${\mathcal{A}}_0$ are countable, $\mathcal{S}$ is countable.

To prove density, start with $f\in L^p(\mu )$ and $\epsilon >0$. By [L1, L2, given, choose, algebra] [L1], choose a finite-support simple function $s={\sum }_{j=1}^m{c}_j{\mathbf{1}}_{E_j}$ with $\mathrm{\parallel }f-s{\mathrm{\parallel }}_p<\epsilon \mathrm{/}2$. For each $j$, [L2] gives $A_j\in {\mathcal{A}}_0$ with $\mu (A_j)<\mathrm{\infty }$ and $\mu (E_j\mathrm{△}{A}_j)$ arbitrarily small, and each coefficient $c_j$ can be approximated by $q_j\in \mathbb{Q}(i)$. The resulting $t:={\sum }_j{q}_j{\mathbf{1}}_{A_j}$ lies in $\mathcal{S}$ and satisfies $\mathrm{\parallel }s-t{\mathrm{\parallel }}_p<\epsilon \mathrm{/}2$.

Then [L3, step 1.1, step 1.2] $\mathrm{\parallel }f-t{\mathrm{\parallel }}_p\le \mathrm{\parallel }f-s{\mathrm{\parallel }}_p+\mathrm{\parallel }s-t{\mathrm{\parallel }}_p<\epsilon \mathrm{.}$ So $\mathcal{S}$ is countable and dense in $L^p(\mu )$; by [L3], the space is separable.