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

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

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 $r>0$.

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

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

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

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