The metric space associated to a measure space

Question

Let $(X, \mathcal{A}, \mu)$ be a measure space such that $\mu(X) < \infty$. We say that two measurable sets $A$ and $B$ are equivalent if $\mu (A \Delta B) = 0$. The equation $$ d(A,B) = \mu (A \Delta B)$$ defines a metric on $\mathcal{A}$ modulo equivalence.

Question: What information if any is encoded in this metric space?

Here is a trivial example: Assume that $X$ is a finite set, $\mathcal{A} = P(X)$ and $\mu$ is the counting measure. We can give $P(X)$ the structure of an undirected graph: Join $A$ and $B$ by an edge if they differ by a point. Then the metric induced on $\mathcal{A}$ by the measure is exactly the graph metric (i.e the distance between two points is the length of the shortest path).

Answer 1

To elaborate on Pietro's comment: if $Y$ is a set, and $S$ is a subset, define the cut metric $d_S$ as the pseudo-metric given by $d_S(y,y')=|1_S(y)-1_S(y')|$, where $1_S$ is the characteristic function of $S$. Let $Cut(Y)$ be the cone generated by cut metrics in the space of non-negative, symmetric kernels on $Y$. If $Y$ is finite (think of $Y$ as your $P(X)$), then a pseudo-metric $d$ on $y$ belongs to $Cut(Y)$ iff there is a map $f:Y\rightarrow L^1$ such that $d(y,y')=\|f_y-f_{y'}\|_1$. See M. Deza and M. Laurent, "Geometry of cuts and metrics", Springer 1997.

Actually we are not very far from the notion of measured wall space, i.e. a Borel measure $\nu$ on the space $P(X)$ (where $X$ is now an arbitrary set) such that, for every $x,x'\in X$, we have $\nu(E_x\Delta E_{x'})<+\infty$, where $E_x$ is the set of subsets through $x$. Then $d_W(x,x')=:\nu(E_x\Delta E_{x'})$ defines a pseudo-metric on $X$, called the wall metric. The relation between measured wall spaces, median metric spaces and embeddability into $L^1$ has been worked out in a nice paper by I. Chatterji, C. Drutu and F. Haglund, see http://fr.arxiv.org/PS_cache/arxiv/pdf/0704/0704.3749v4.pdf

Answer 2

This metric recovers the measure space up to measure-preserving transformations. Fix a point to be $0$. You can take unions and intersections relative to that point, using only the metric. For instance, $X\cap Y$ is the point farthest from $0$ such that two triangle inequalities are exact: $d(X,P)+d(P,0)=d(X,0)$ and $d(Y,P)+d(P,0)=d(Y,0)$. Similarly for anything else you want to do with sets. So you get the entire sigma-algebra structure, modulo sets of measure $0$.

However, that structure isn't very much. Any measure-preserving transformation between two measure spaces is an isomorphism from this perspective.

Answer 3

Complementing the previous answer and comment: These metric is used in the context of shape optimization problems (in the case of $X$ a subset of $\mathbb{R}^d$ with the Lebesgue measure). In "Shapes and Geometries" by Delfour and Zolesio and you can find several interesting results in that book. E.g. the $L^p$-norm can be used ($1\leq p < \infty$) instead of $L^1$ in Pietro's comment and lead to the same topology, you can approximate any Lebesgue measurable set by $C^\infty$-domains in these metrics, the characteristic functions of convex sets form a closed subset.

Answer 4

The measure space can be infinite--you just need to restrict to the sets of finite measure.

Pietro pointed out that the resulting metric space embeds isometrically into $L_1(\mu)$. Conversely, $L_1(\mu)$ embeds isometrically into the sets of finite measure in the measure space $(X\times \Bbb{R}, \mathcal{A}, \mu \times m)$ ($m$ is Lebesgue measure)--map the function $f$ to the region under its graph. As Alain points out, you can then snowflake into $L_2$. This is an easy way of seeing that $L_1$ with the snowflake distance $d(f,g) := \|f-\|g_2^{1/2}$ embeds isometrically into $L_2$, and hence $L_1$ uniformly embeds into $L_2$.