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Let $X$ be our space and $\wp(X)$ its power set. Then $\rho:\wp(X)\times\wp(X)\to [0,\infty]$ defined by $\rho(A,B)=m^{*}(A\Delta B)$ is a pseudo metric on $\wp(X)$ (the possibility $\rho(A,B)=\infty$ does not seem to be a problem).

We might restrict our attention to the $\sigma$-algebra of measurable sets, say $\mathcal{M}$, and assume $m(X)<\infty$. Then $\rho$ definitely gives a topology on $\mathcal{M}$. Moreover, it is readily seen that if $\mathcal{A}\subset\wp(X)$ generates $\mathcal{M}$, then $\mathcal{A}$ is dense in $\mathcal{M}$ in this topology.

It seems to me that such a metric might be interesting but somehow I cannot find any book even mentioning this topology. Thus I wonder whether someone has looked into this.

Any reference would be welcome! Thanks!

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It is mentioned in the textbook of Halmos on measure theory. – Michael Greinecker Feb 11 '12 at 8:17
up vote 2 down vote accepted

You’ll find some information about the pseudometric and a couple of references in the answers to this earlier question. The metric induced on the quotient if you mod out the null sets is apparently known as the symmetric difference metric, the Fréchet-Nikodym-Aronszyan (or simply Fréchet-Nikodym) metric, or the measure metric.

One way in which it is used can be seen in this post to Terry Tao’s blog.

A generalization of this pseudometric is treated in Extension of measures and integrals by the help of a pseudometric by Beloslav Riečan (Mathematica Slovaca, Vol. 27 (1977), No. 2, 143-152); it looks interesting at first glance, but I’ve not actually read the paper.

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@Hui Yu: You can find some applications of (and references to) this metric in my answer at – Dave L. Renfro Feb 13 '12 at 15:16

Note that $\rho (A,B) = \int_X | \chi_A - \chi_B | dm$, so $\rho$ is just the $L^1$ metric restricted to the subset of indicator functions. So you can already deduce some properties of this metric space by using what you already know about $L^1$.

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