TITLE: The metric space associated to a measure space
QUESTION [10 upvotes]: Let $(X, \mathcal{A}, \mu)$ be a measure space such that $\mu(X) < \infty$. We say that two measurable sets $A$ and $B$ are equivalent if $\mu (A \Delta B) = 0$. The equation $$ d(A,B) = \mu (A \Delta B)$$ defines a metric on $\mathcal{A}$ modulo equivalence. 

Question: What information if any is encoded in this metric space? 

Here is a trivial example: Assume that $X$ is a finite set, $\mathcal{A} = P(X)$ and $\mu$ is the counting measure. We can give $P(X)$ the structure of an undirected graph: Join $A$ and $B$ by an edge if they differ by a point. Then the metric induced on $\mathcal{A}$ by the measure is exactly the graph metric (i.e the distance between two points is the length of the shortest path).

REPLY [7 votes]: This metric recovers the measure space up to measure-preserving transformations. Fix a point to be $0$. You can take unions and intersections relative to that point, using only the metric. For instance, $X\cap Y$ is the point farthest from $0$ such that two triangle inequalities are exact: $d(X,P)+d(P,0)=d(X,0)$ and $d(Y,P)+d(P,0)=d(Y,0)$. Similarly for anything else you want to do with sets. So you get the entire sigma-algebra structure, modulo sets of measure $0$.
However, that structure isn't very much. Any measure-preserving transformation between two measure spaces is an isomorphism from this perspective.