Let $(X,\mathscr{A},\mu)$ be a probability space and let $\{A_1,\ldots,\}\subset\mathscr{A}$ be a countable family of sets with small measure: say $\mu(A_i)\le\epsilon$. I am trying to show that one can find a countable (disjoint!) partition $\{B_i\}$ of $X$ with the following property: Each $A_i$ is covered by some $(B_j)_{j\in J}$ such that $|J|$ is small (say, $1/\epsilon$) and $\mu(\cup_{j\in J}B_j)$ is not too large (say, $O(\epsilon)$ or even $O(\sqrt\epsilon)$).

We can assume that $\mathscr{A}$ is a Borel $\sigma$-algebra induced by some metric, if it helps.

Edit. It was pointed out by Fedja and others that the previous formulation, which required a condition like $\mu(B_i)\le\epsilon^2$, has atomic counterexamples.