the pseudometric induced by a measure Let $(X, \Sigma, \mu)$ be a measure space.
We can define a pseudometric $d$ on $\Sigma$ in the following way:
$$d(A, B) = \mu(A\bigtriangleup{}B)$$
where $A\bigtriangleup{}B = (A\cup{}B)\setminus{}(A\cap{}B)$ is the symmetric difference.
Does this metric have a name? What does the topology induced by $d$ on $\Sigma$ tell us about the measure?
 A: Assuming $\mu(X)<\infty$, this is the restriction of the $L^1$ metric on $L^1(X)$ to the subspace consisting of characteristic (a.k.a. indicator) functions.  Clearly this is a closed (topological) subspace, and therefore $\Sigma$ is a complete metric space under this metric.
As far as I know, this metric itself doesn't have a name.  In some sense, it should give precisely the same information about the measure that you can get from the space of $L^1$ functions.
Edit: As Theo pointed out, the last sentence isn't quite correct, since the subspace of characteristic functions cannot actually be recognized from within $L^1$.  For example, $L^1([0,1])$ and $L^1([0,2])$ are isomorphically isometric via the map $f(x) \mapsto f(x/2)/2$, but the corresponding $\Sigma$'s are not isometric. In general, the total measure $\mu(X)$ of the underlying space cannot necessarily be recovered from $L^1(X)$, but $\mu(X)$ is the diameter of $\Sigma$ as a metric space.
Edit #2:  Some further comments:


*

*Note that $\Sigma$ is a topological group under $\bigtriangleup$, and is therefore homogeneous.  As a result, we can assume that we know which point of $\Sigma$ represents the empty set $\emptyset$.

*Let $\sim$ be the equivalence relation
$$
A\sim B \qquad\Leftrightarrow\qquad \mu(A \bigtriangleup B) = 0.
$$
Since $\Sigma$ is homogeneous, every equivalence class under $\sim$ has the same cardinality as the equivalence class of $\emptyset$.  Thus, we can use the pseudometric structure on $\Sigma$ to determine the cardinality of the collection of measure-zero sets.

*Assuming $\mu(X) < \infty$, the metric structure on $\Sigma$ can also be used to reconstruct the $\sigma$-algebra structure on $\Sigma/\sim$.  In particular, if $A,B\in \Sigma/\sim$, we can tell whether $A\subset B$ (modulo sets of measure zero) by checking whether
$$
d(\emptyset,A) + d(A,B) = d(\emptyset,B)\text{,}
$$
and this lets us reconstruct the notions of countable union and complement.

*According to point #3, the metric structure on $\Sigma$ lets us reconstruct $\Sigma/\sim$ as a measure algebra (see this reference for the definitions used in this paragraph).  Moreover, it should be clear that $\Sigma/\sim$ and $\Sigma'/\sim$ are conjugate as measure algebras if and only if they are isometric as metric spaces.  
We conclude that, in the case where $\mu(X) < \infty$, the metric structure on $\Sigma$ contains only the following information:
(1) The cardinality of the collection of sets of measure zero and
(2) The conjugacy class of $(X,\Sigma,\mu)$ as a measure algebra.
According to this reference, any two separable, non-atomic probability spaces are measure algebra conjugate.  Therefore, as long as $(X,\Sigma,\mu)$ is separable and non-atomic, the only information contained in point (2) is the total measure $\mu(X)$ of $X$.
A: The function $d$ is sometimes called the Fréchet-Nikodym metric; for example, see section 1.12(iii) on page 53 of Measure Theory: Volume 1 by V. Bogachev. 

Added: Google Books led me to the Encyclopedia of Distances by Michel M. Deza, Elena Deza. There the metric is called the symmetric difference metric, or the Fréchet-Nikodym-Aronszyan metric, or the measure metric.   
A: Separability of this spaces gives separability of the $L^p$ spaces, $1\le p < \infty$.
