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Let $(X, \Sigma, \mu)$ be a measure space.

We can define a pseudometric $d$ on $\Sigma$ in the following way:

$d(A, B) = \mu(A\bigtriangleup{}B)$

where $A\bigtriangleup{}B = (A\cup{}B)\setminus{}(A\cap{}B)$ is the symmetric difference.

Does this metric have a name? What does the topology induced by $d$ on $\Sigma$ tell us about the measure?

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    It's just the standard (pseudo)metric induced by the measure; normally this is defined on the quotient measure algebra (modding out the null sets), making it a metric on that. It's used to define a separable measure: iff this metric space is separable.2011-06-08

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Assuming $\mu(X)<\infty$, this is the restriction of the $L^1$ metric on $L^1(X)$ to the subspace consisting of characteristic (a.k.a. indicator) functions. Clearly this is a closed (topological) subspace, and therefore $\Sigma$ is a complete metric space under this metric.

As far as I know, this metric itself doesn't have a name. In some sense, it should give precisely the same information about the measure that you can get from the space of $L^1$ functions.

Edit: As Theo pointed out, the last sentence isn't quite correct, since the subspace of characteristic functions cannot actually be recognized from within $L^1$. For example, $L^1([0,1])$ and $L^1([0,2])$ are isomorphically isometric via the map $f(x) \mapsto f(x/2)/2$, but the corresponding $\Sigma$'s are not isometric. In general, the total measure $\mu(X)$ of the underlying space cannot necessarily be recovered from $L^1(X)$, but $\mu(X)$ is the diameter of $\Sigma$ as a metric space.

Edit #2: Some further comments:

  1. Note that $\Sigma$ is a topological group under $\bigtriangleup$, and is therefore homogeneous. As a result, we can assume that we know which point of $\Sigma$ represents the empty set $\emptyset$.

  2. Let $\sim$ be the equivalence relation $ A\sim B \qquad\Leftrightarrow\qquad \mu(A \bigtriangleup B) = 0. $ Since $\Sigma$ is homogeneous, every equivalence class under $\sim$ has the same cardinality as the equivalence class of $\emptyset$. Thus, we can use the pseudometric structure on $\Sigma$ to determine the cardinality of the collection of measure-zero sets.

  3. Assuming $\mu(X) < \infty$, the metric structure on $\Sigma$ can also be used to reconstruct the $\sigma$-algebra structure on $\Sigma/\sim$. In particular, if $A,B\in \Sigma/\sim$, we can tell whether $A\subset B$ (modulo sets of measure zero) by checking whether $ d(\emptyset,A) + d(A,B) = d(\emptyset,B)\text{,} $ and this lets us reconstruct the notions of countable union and complement.

  4. According to point #3, the metric structure on $\Sigma$ lets us reconstruct $\Sigma/\sim$ as a measure algebra (see this reference for the definitions used in this paragraph). Moreover, it should be clear that $\Sigma/\sim$ and \Sigma'/\sim are conjugate as measure algebras if and only if they are isometric as metric spaces.

We conclude that, in the case where $\mu(X) < \infty$, the metric structure on $\Sigma$ contains only the following information:

(1) The cardinality of the collection of sets of measure zero and

(2) The conjugacy class of $(X,\Sigma,\mu)$ as a measure algebra.

According to this reference, any two separable, non-atomic probability spaces are measure algebra conjugate. Therefore, as long as $(X,\Sigma,\mu)$ is separable and non-atomic, the only information contained in point (2) is the total measure $\mu(X)$ of $X$.

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    Your addition is very nice and shows that in the separable case my point is essentially moot.2011-06-09
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The function $d$ is sometimes called the Fréchet-Nikodym metric; for example, see section 1.12(iii) on page 53 of Measure Theory: Volume 1 by V. Bogachev.


Added: Google Books led me to the Encyclopedia of Distances by Michel M. Deza, Elena Deza. There the metric is called the symmetric difference metric, or the Fréchet-Nikodym-Aronszyan metric, or the measure metric.

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    Thanks! That's what I feared... So I'll have to dig in the library if I really want to know. I'll post a comment here if I should happen to find something worth mentioning.2011-06-09
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Separability of this spaces gives separability of the $L^p$ spaces, $1\le p < \infty$.