Correct condition for $L_p$ to be separable.

Question

I am a physics undergraduate so I only had two lectures about measure theory. Could someone explain to me what is the most correct statement between these two ($p<\infty$):

1. From Is $L^p$ separable?. Let $(X,\sigma,\mu)$ be a measure space where $\mu$ is a sigma finite measure. If there exists a succession $A=A_1,...,A_n,... ⊂ \sigma$ such that, for any $>0$ and $M∈Σ$ one can find $A_k∈A$ with $(_k△M)<$ then the metric space $(L_p,||\cdot||_p)$ is separable.
2. From Countably generated $\sigma$-algebra implies separability of $L^p$ spaces. Let $(X,\sigma,\mu)$ be a measure space where $\mu$ is a sigma finite measure. If the sigma algebra $\sigma$ is countably generated (i.e. we can find a succession $A=A_1,...A_n,...$ such that $\sigma$ is generated by $A$) then the metric space $(L_p,||\cdot||_p)$ is separable.