Every complete doubling metric space carries a doubling measure

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 Rⁿ carries a measure of homogeneity exponent n n . These results are based on the case of compact X X due to Vol ′ berg and Konyagin.