On measures with the doubling condition

The author characterizes metric spaces $(X,\rho)$ that carry a nontrivial measure $\mu$ with the doubling condition
$$ \forall\,x\in X,\quad R>0\qquad \mu(B(x,2R))\leqslant C\mu(B(x,R)), $$
where $B(x,R)=\{y:\rho(x,y)\leqslant R\}$. In particular, such a measure exists on any compact set in $\mathbf R^n$.
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