Consider a finite measure space $(X,\Sigma,\mu)$.
Consider the function $d:\Sigma \times \Sigma \to [0,1]$ given by
$$d(\sigma\_1,\sigma\_2) = \mu \left\{ (\sigma\_1^c \cap \sigma\_2) \cup (\sigma\_1 \cap \sigma^c\_2) \right\}.$$ One can verify that $d$ is a pseudometric (where $d(\sigma\_1,\sigma\_2) = 0$ means that $\sigma\_1$ and $\sigma\_2$ differ by a set of measure zero). Does $d$ have a name?



Is there a name for this metric on a Borel sets
The book [Dictionary of distances](https://books.google.com/books?id=I-PQH8gcOjUC&lpg=PA25&ots=5bY662ttFI&dq=counting%20symmetric%20difference%20metric&pg=PA25#v=onepage&q=counting%20symmetric%20difference%20metric&f=false) by Deza and Deza lists several names for this object (and its induce metric on the quotient when we identify two sets that differ by measure zero).


* Symmetric difference (pseudo)metric
* Frechet-Nikodym-Aronszayn distance
* Measure metric




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Google also tells me that [Dave Renfro has a post on Sci.Math](https://groups.google.com/forum/#!msg/sci.math/xR-j_nLoFQE/7n1xXggU4OwJ) which includes a mini survey of some of the research problems where this metric comes into play. 


