# Pseudo metric defined in term of outer measure

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

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 $L^1$.