I would like to better understand the following definition:
$(M, \mathcal {A}, \mu) $ a probability space is separable if there exists a countable family $ \mathcal {E} \subset \mathcal {A} $ such that for all $ A \in \mathcal{A} $ and $ \varepsilon> $ 0, there is $ B \in \mathcal {E} $ such that $ \mu (A \triangle B )<\varepsilon$.
I wonder if the Lebesgue probability space ($M=[0,1],$ $\mathcal {A}=$Lebesgue measurable sets in [0,1], $\mu=m$) is separable, as finding $\mathcal {E} $?
Thanks for any suggestions