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.


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