Every complete doubling metric space carries a doubling measure

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.