The metric space associated to a measure space - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T10:47:31Z http://mathoverflow.net/feeds/question/80026 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/80026/the-metric-space-associated-to-a-measure-space The metric space associated to a measure space Daniel Barter 2011-11-04T09:31:11Z 2011-11-04T18:18:04Z <p>Let $(X, \mathcal{A}, \mu)$ be a measure space such that $\mu(X) &lt; \infty$. 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. </p> <blockquote> <p>Question: What information if any is encoded in this metric space? </p> </blockquote> <p>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). </p> http://mathoverflow.net/questions/80026/the-metric-space-associated-to-a-measure-space/80034#80034 Answer by Alain Valette for The metric space associated to a measure space Alain Valette 2011-11-04T10:57:12Z 2011-11-04T10:57:12Z <p>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. </p> <p>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 $\nu(E_x\Delta E_{x'})&lt;+\infty$, 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 <a href="http://fr.arxiv.org/PS_cache/arxiv/pdf/0704/0704.3749v4.pdf" rel="nofollow">http://fr.arxiv.org/PS_cache/arxiv/pdf/0704/0704.3749v4.pdf</a></p> http://mathoverflow.net/questions/80026/the-metric-space-associated-to-a-measure-space/80042#80042 Answer by Dirk for The metric space associated to a measure space Dirk 2011-11-04T12:20:24Z 2011-11-04T12:20:24Z <p>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 ($1\leq p &lt; \infty$) 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.</p> http://mathoverflow.net/questions/80026/the-metric-space-associated-to-a-measure-space/80065#80065 Answer by Bill Johnson for The metric space associated to a measure space Bill Johnson 2011-11-04T16:23:55Z 2011-11-04T16:23:55Z <p>The measure space can be infinite--you just need to restrict to the sets of finite measure.</p> <p>Pietro pointed out that the resulting metric space embeds isometrically into <code>$L_1(\mu)$</code>. Conversely, <code>$L_1(\mu)$</code> embeds isometrically into the sets of finite measure in the measure space <code>$(X\times \Bbb{R}, \mathcal{A}, \mu \times m)$</code> ($m$ 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 <code>$d(f,g) := \|f-\|g_2^{1/2}$</code> embeds isometrically into $L_2$, and hence $L_1$ uniformly embeds into $L_2$. </p> http://mathoverflow.net/questions/80026/the-metric-space-associated-to-a-measure-space/80078#80078 Answer by Will Sawin for The metric space associated to a measure space Will Sawin 2011-11-04T18:18:04Z 2011-11-04T18:18:04Z <p>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 $0$.</p> <p>However, that structure isn't very much. Any measure-preserving transformation between two measure spaces is an isomorphism from this perspective.</p>