Doubling measure

Summary

A doubling measure on a metric space is a Borel measure with the following property, for a fixed constant $C$. Doubling the radius of any ball multiplies its measure by at most $C$. Lebesgue measure on $\R^n$ is the model example, with constant $2^n$. Many tools of real analysis on $\R^n$ extend to metric spaces with a doubling measure. Examples are the Vitali covering lemma, maximal function bounds and the Lebesgue differentiation theorem. A metric space that carries a doubling measure is a doubling metric space.