Let $I=[0,1]$, $\mathcal{A}$ is the family that contains all the Lebesgue measurable sets of $I$, for any $A_1,A_2\in \mathcal{A}$, we define a metric $$d(A_1,A_2)=\int |1_{A_1}-1_{A_2}|.$$
A $\varepsilon$-net of $\mathcal{A}$ is a sub-family $\mathcal{A}'\subset \mathcal{A}$ such that $$\forall A\in \mathcal{A},\exists A'\in \mathcal{A}',~s.t.~d(A,A')\le \varepsilon.$$
My question is:
For all $\varepsilon>0$, does there exist a countable $\varepsilon$-net for $\mathcal{A}$?
To make $d$ a metric you have to think of your set as a subset of $L^{1}$ by identifying sets which differ only by a null set. In that case the assertion is true. Consider finite disjoint unions of intervals with rational end points and their characteristic functions. Characteristic function of any $A \in\mathcal A$ can be approximated in $L^{1}$ norm by those of finite disjoint unions of intervals and then we can approximate these intervals by those with rational end points which proves the assertion.