# Is there a name for this metric on a Borel sets

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?

• It is also the distance between the characteristic or indicator functions of the two sets in the Banach space $L^1(\mu)$. – oeiras Mar 17 '16 at 18:35
Since $(\sigma_1^c \cap \sigma_2) \cup (\sigma_1 \cap \sigma^c_2)$ is simply the symmetric difference of the sets $\sigma_1, \sigma_2$ (also denoted more concisely as $\sigma_1 \triangle \sigma_2$), your metric is often called the symmetric difference (pseudo)metric induced by the measure $\mu$.