An alternative way to get the Lebesgue $\sigma $-algebra $\mathcal{L} $ from the Borel algebra $B$ is to set $E\sim J$ iff $d(E,J):=\lambda(E\Delta J)=0$ for $E,J\in B$. Then the completion of $(X,d)$ as a metric space, where $X=B/\sim$, is equivalent to completion of $B$.
My question is: Are there known generalizations? Given a measure space $(X,\Sigma ,\mu)$, what are the conditions on $X$ and $\Sigma$ so that the completion of the measure space is equivalent to the completion of $M:=\Sigma/\sim$ as a metric space, with $\sim $ as above? Do we have to use the symmetric difference to define $d$ on $M$?