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?
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2 Answers
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The book Dictionary of distances 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
Google also tells me that Dave Renfro has a post on Sci.Math which includes a mini survey of some of the research problems where this metric comes into play.
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3$\begingroup$ It is also the distance between the characteristic or indicator functions of the two sets in the Banach space $L^1(\mu)$. $\endgroup$– oeirasCommented Mar 17, 2016 at 18:35
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$\begingroup$ As a matter of historical interest, Saks used this space to prove what became known as the Vitali-Hahn-Saks theorem on the limit of sequences of measures by applying Baire's category theorem to it as a closed subset of a Banach space. $\endgroup$– oeirasCommented Mar 18, 2016 at 7:10
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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 $\mu$.
answered Mar 17, 2016 at 16:35