2
$\begingroup$

I'm looking for examples of spaces $X$ such that:

  1. $X$ is a probability space.
  2. $X$ is a metric space.
  3. If $x \in X$ and $0 < r \leq R$ then $\frac{Pr(B(x,r))}{Pr(B(x,R))} \geq \frac{r}{R}$.

I already have some examples:

  1. A segment $[a,b] \subset \mathbb{R}$ with the natural probability and metric.
  2. Generalization of $1$: A finite tree with positive weights on the edges (where the weights sum up to 1, each edge $e$ is equivalent to a segment $[0,weight(e)]$ and the entire space is a quotient of the disjoint union of the edges which identifies the ends of the edges according to the structure of the tree). See Didier's comment.
  • 0
    Here's a fancier version of the above example. (I hope this works; I cannot trust my calculations yet.) Take a unit interval, and a circle of circumference $2$. Identify one endpoint of the interval with one of the points in the circle. The metric is the natural metric. The probability density is uniform over the interval and over the circle individually, and the total probability of the interval (or the circle) is $1/2$. (Note that since the circumference of the circle is twice the length of the interval, the density over the interval is double that over the circle.)2011-08-01

0 Answers 0