Countably generated $\sigma$-algebra implies separability of $L^p$ spaces Let $\Sigma = \sigma(\mathcal C)$ be the $\sigma$-algebra generated by the countable collection of sets $\mathcal C \subset \mathcal{P}(X)$. How can I prove that if $\mu$ is a $\sigma$-finite measure on $(X,\Sigma)$ then $L^p(X)$ is separable for $1 \le p < \infty$?
I know that simple functions are dense in $L^p(X)$, so I would like to find a countable subset of the set of simple functions that is dense in them. Could you help me please?

Begin as follows. You have a countable collection of sets generating the algebra.  Now take the finite intersections of all elements of this collection; it is countable.  Now take the finite unions of the countable family of finite intersections.  This is an algebra of sets; the smallest $\sigma$-algebra generating it is $\Sigma$; let us denote this algebra by $\mathcal{A}$.
The simple functions 
$$\mathcal{S} = \left\{\sum_{k=1}^n c_k \chi_{A_k}, A_1, A_2, \cdots A_k \in \mathcal{A}, 
c_1, c_2, .... c_n\in \mathbb{Q}, n\in \mathbb{N}\right\}$$
constitute a countable set.  
Choose $E\in \sigma$.  For any $\epsilon > 0$ you can choose $A\in \mathcal{A}$ 
so that $\mu(A\Delta E) < \epsilon$.  This says that every characteristic function is in the closure of $\mathcal{S}$.  Can you continue and show that $\mathcal{S}$ is dense in $\mathcal{L}^p$?
