I'm looking for examples of spaces $X$ such that:
- $X$ is a probability space.
- $X$ is a metric space.
- 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:
- A segment $[a,b] \subset \mathbb{R}$ with the natural probability and metric.
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.