Abstract
We prove that a complete metric space X X carries a doubling measure if and only if X X is doubling and that more precisely the infima of the homogeneity exponents of the doubling measures on X X and of the homogeneity exponents of X X are equal. We also show that a closed subset X X of R n \mathbf {R}^{n} carries a measure of homogeneity exponent n n . These results are based on the case of compact X X due to Vol ′ ^{\prime } berg and Konyagin.
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