J. Luukkainen et E. Saksman, EVERY COMPLETE DOUBLING METRIC SPACE CARRIES A DOUBLING MEASURE, Proceedings of the American Mathematical Society, 126(2), 1998, pp. 531-534
We prove that a complete metric space X carries a doubling measure if
and only if X is doubling and that more precisely the infima of the ho
mogeneity exponents of the doubling measures on X and of the homogenei
ty exponents of X are equal. We also show that a closed subset X of R-
n carries a measure of homogeneity exponent n. These results are based
on the case of compact X due to Vol'berg and Konyagin.