Hausdorff dimension and doubling measures on metric spaces

Vol$'$berg and Konyagin have proved that a compact metric space carries a nontrivial doubling measure if and only if it has finite uniform metric dimension. Their construction of doubling measures requires infinitely many adjustments. We give a simpler and more direct construction, and also prove that for any $\alpha > 0$, the doubling measure may be chosen to have full measure on a set of Hausdorff dimension at most $\alpha$.