Can the Borel sets be recovered from the Lebesgue sets and the null sets?

Suppose that one is given a set $$X$$ equipped with a pair of $$\sigma$$-algebras $$\mathcal{L}$$ and $$\mathcal{N}$$. Suppose that $$\mathcal{L}$$ and $$\mathcal{N}$$ came from putting a nice topology on $$X$$, then putting a nice measure on the Borel sets $$\mathcal{B}$$ of $$X$$, then letting $$\mathcal{L}$$ be the completion of $$\mathcal{B}$$, then letting $$\mathcal{N}$$ be the measure-zero sets in $$\mathcal{L}$$, then forgetting the measure, then forgetting the topology. Is there any hope that one could recover $$\mathcal{B}$$ from $$\mathcal{L}$$ and $$\mathcal{N}$$? If the topology is not assumed to be nice, then the answer is no, because for example the set of all usual Lebesgue subsets of $$\mathbb{R}$$ is the set of all Borel subsets of $$\mathbb{R}$$ when $$\mathbb{R}$$ is equipped with the weird topology referenced here.