Metric defined on a measure space by $d(A,B)=\mu (A\Delta B)$? Yesterday I suddenly came up with this idea: let $(X,\Sigma,\mu)$ be a measure space. For $A,B\in \Sigma$, define
$$
d(A,B)=\mu(A\Delta B), A\Delta B=(A\cup B)\backslash (A\cap B).
$$
By careful consideration on a Venn diagram, one can show that $d$ satisfy the triangular inequality, and is hence a pseudo-metric. If we identify all sets that have distance $0$ from each other, we get a metric space $\overline \Sigma$.
Is there any application of this metric in mathematics? Can this theory be developed any further?
 A: This answer is an elaboration of what has been written already in the comments.
Under the extra assumption that $\mu(X)<\infty$, your metric space $(\overline\Sigma,d)$ admits an isometric embedding into the metric space $L^1(X,\Sigma,\mu)$. The embedding is given explicitly by sending an equivalence class $[A]\in\overline\Sigma$ to the equivalence class $[1_A]\in L^1(X,\Sigma,\mu)$ containing the indicator function $1_A$. (Recall that elements of the $L^p$ spaces are equivalence classes of measurable functions which agree up to $\mu$-null sets, and that $1_A$ denotes the function taking the value $1$ on the set $A$ and taking the value $0$ on its complement.) This embedding is well-defined (i.e., not multi-valued) since the functions $1_A$ and $1_B$ are equal on $(A\triangle B)^c$, thus if $d(A,B)=0$ then $\mu(A\triangle B)=0$ so $1_A$ and $1_B$ agree up to a $\mu$-null set. The embedding is an isometry since (as noted in the comments)
$$
\int_X |1_A-1_B|\ d\mu=\int_X 1_{A\triangle B}\ d\mu=\mu(A\triangle B)=d(A,B).
$$
(The identity $|1_A-1_B|=1_{A\triangle B}$ is the key to all of this. It also shows that the triangle inequality for $d(A,B)$ is actually a special case of the usual triangle inequality for functions, in the form $|1_A-1_C|\leq |1_A-1_B|+|1_B-1_C|$.)
The image of this embedding can be characterized as the set of equivalence classes in $L^1(X,\Sigma,\mu)$ which contain a function taking values in $\{0,1\}$. Thus, the study of the properties of your metric space amounts to the study of a (relatively simple to describe) subset of the (very well-studied and famous) metric space $L^1(X,\Sigma,\mu)$.
By the way, this phenomenon (embedding a space of sets into a space of functions) works in lots of different settings and is formalized by the notion of subobject classifier, which is the role played by the set $\{0,1\}$ in the previous paragraph.
