Let $(E,\mathcal E,\mu)$ be a probability space.
Under which further assumption are we able to show that $L^2(\mu)$ is separable?
Most lecture books on measure theory only show separability of $L^2(\mu)$ for $(E,\mathcal E,\mu)=\left(\mathbb R^d,\mathcal B(\mathbb R)^{\otimes d},\lambda^{\otimes d}\right)$, where $\lambda$ is the Lebesgue measure on $\mathcal B(\mathbb R)$.
I guess (but I'm not sure) that we need something like separability of $\mathcal E$ equipped with the metric $$d_\mu(A,B):=\mu(A\Delta B)\;\;\;\text{for }A,B\in\mathcal E,$$ where $\Delta$ denotes the symmetric difference.
Remark: I know there are several questions regarding the separability of $L^p$-spaces on the board, but almost all of them are concered with $E=\mathbb R^d$.