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 homogeneity exponents of the doubling measures on X and of the homogeneity 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. [References: 11]
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