EVERY COMPLETE DOUBLING METRIC SPACE CARRIES A DOUBLING MEASURE

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.