Bounded ball measure on compact metric space

Question

Fix $c>1$. Let $(X,d)$ be a separable compact metric space, does there necessarily exist a Borel probability measure $\nu$ on $(X,d)$ such that

• $\operatorname{sup}_{x \in X,r>0}\frac{\nu(\mathrm{Ball}(x,cr))}{\nu(\mathrm{Ball}(x,r))}<\infty$,
• $\nu(\mathrm{Ball}(x,r))>0$ for every $x \in X$ and $r>0$.

Answer 1

Are you asking if there is a $c>1$ and a measure $\nu$ with this property? And is the first condition supposed to hold with a uniform upper bound? (If not, any probability measure satisfying the second condition would work, as Steve points out.)

If so, you are asking for the existence of a non-trivial "doubling measure". A necessary and sufficient condition for a complete metric space to carry a non-trivial doubling measure is that it is a "doubling metric space".

A metric space is a "doubling metric space" if every ball $B(x,r)$ can be covered by $N$ balls of radius $r/2$, where $N$ is a fixed constant. See wikipedia: https://en.wikipedia.org/wiki/Doubling_space

If a complete metric space carries a non-trivial doubling measure, it must be a doubling metric space, by a fairly standard covering argument. (See, for example, Heinonen's book Lectures on Analysis on Metric Spaces.)

The reverse implication, that every complete doubling metric space carries a non-trivial doubling measure, is more difficult. It was proven by Vol'berg-Konyagin in the compact case and Luukkainen-Saksman in the general case. See Luukkainen-Saksman (https://www.ams.org/journals/proc/1998-126-02/S0002-9939-98-04201-4/S0002-9939-98-04201-4.pdf) for the details and references.