Existence of doubling measures via generalised nested cubes (Q2845753)

A metric space \(X\) is doubling if every closed ball of radius \(2r\) (\(B(x, 2r)\), \(x \in X\)) can be covered by finitely many balls of radius \(r\), with the number of such \(r\)-balls having an upper bound independent of \(x\) and \(r\). A Borel regular outer measure \(\mu\) on a metric space is doubling if there exists a constant \(D\), \(1 \leq D < \infty\), such that \(0< \mu(B(x,2r)) \leq D \mu(B(x,r))<\infty\). In a complete doubling metric space, the authors construct a nested family of ``cubes'' that share most of the desirable properties of dyadic cubes in Euclidean spaces, adapt an ultrametric on the ``codes'' (certain indices of the ``cubes''), and obtain a doubling measure. The proof uses nested partitions and mass distribution principles, allowing the authors to directly obtain the result for the unbounded case. As an application, they show that for each \(\epsilon > 0\), there exists a doubling measure having full measure on a set of packing dimension at most \(\epsilon\), a result previously known only for the Hausdorff dimension (e.g. [\textit{J.-M. Wu}, Proc. Am. Math. Soc. 126, No. 5, 1453--1459 (1998; Zbl 0897.28008)]).