Every complete doubling metric space carries a doubling measure

By J. Luukkainen, Eero Saksman. 1998

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k-NN Regression Adapts to Local Intrinsic Dimension (2011, background): “nd dsuch that for any x∈ X, for any r≥ 0 and any 0 <ǫ<1, we have µ(B(x,r)) ≤ Cǫ−dµ(B(x,ǫr)). Here dacts as a dimension. It is not hard to show that dcan be chosen as log2C dband Cas C (see e.g. [14]). A simple example of a doublingmeasure is the Lebesgue volume in the Euclidean space Rd. For a”
A framework for non-homogeneous analysis on metric spaces, and the RBMO space of Tolsa (2009, background): “Conversely, if (X, d) is a complete, geometrically doubling metric space, then there exists a Borel measure μ on X such that (X, d, μ) is measure doubling [13], [22], [23].”
Fast construction of nets in low dimensional metrics, and their applications (2004, background): “ty that for every x∈Pand r>0, the ratio µ(b(x,2r))/µ(b(x,r)) is bounded, where b(x,r) = {y: d(x,y) ≤r}. Vol′berg and Konyagin [VK87] proved that for ﬁnite metrics (and in fact for complete metrics [LS98]) the existence of doubling measure is quantitatively equivalent to the metric being doubling.”
A framework for non-homogeneous analysis on metric spaces, and the RBMO space of Tolsa (2009, background): “Conversely, if (X, d) is a complete, geometrically doubling metric space, then there exists a Borel measure μ on X such that (X, d, μ) is measure doubling [13], [22], [23].”
Sobolev Mappings between Manifolds and Metric Spaces (2009, methodology): “The following beautiful characterization of metric spaces supporting doubling measures was proved by Volberg and Konyagin [62, 82].”
Assouad type dimensions and homogeneity of fractals (2013, background): “n sets themselves. Although their denitions depend only on the structure of the set, the Assouad and lower dimensions have a fascinating link with certain classes of measures. Luukkainen and Saksman [LuS] (see also [KV]) proved that the Assouad dimension of a compact metric space Xis the inmum of s&”
Geometric embeddings of metric spaces (2003, unknown): “Conversely, if X is a complete doubling space, then X carries a nontrivial doubling measure [77], [51].”
The Hardy space H1 on non-homogeneous metric spaces (2010, background): “Conversely, if (X , d) is a complete geometrically doubling metric space, then there exists a Borel measure μ on X such that (X , d, μ) is a space of homogeneous type; see [10] and [18].”
Random Martingales and localization of maximal inequalities (2009, background): “It is a classical fact, due to Vol′berg and Konyagin [54] in the case of compact spaces, and Luukkainen and Saksman [33] in the case of general complete spaces (see also [55] and chapter 13 in [27]), that X admits a non-degenerate measure µ which is doubling with constant λ2 (the power 2 can be repl”
Stability of elliptic Harnack inequality (2016, methodology): “tants C0 >1, 0 <β1 ≤ β2 and a measure µon X which is (C0,β1,β2)-capacity good. We begin by adapting the argument in [VK] to measure with the desired properties in a ball B(x0,R). We then follow [LuS] and construct µas a weak∗ limit of measures deﬁned on a family of balls. {P:ballmeas} Proposit”