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# Arkiv för Matematik

## Volume 58 (2020)

### Number 2

### The doubling metric and doubling measures

Pages: 243 – 266

DOI: https://dx.doi.org/10.4310/ARKIV.2020.v58.n2.a2

#### Authors

#### Abstract

We introduce the so-called doubling metric on the collection of non-empty bounded open subsets of a metric space. Given an open subset $\mathbb{U}$ of a metric space $X$, the predecessor $\mathbb{U}_\ast$ of $\mathbb{U}$ is defined by doubling the radii of all open balls contained inside $\mathbb{U}$, and taking their union. The predecessor of $\mathbb{U}$ is an open set containing $\mathbb{U}$. The directed doubling distance between $\mathbb{U}$ and another subset $\mathbb{V}$ is the number of times that the predecessor operation needs to be applied to $\mathbb{U}$ to obtain a set that contains $\mathbb{V}$. Finally, the doubling distance between open sets $\mathbb{U}$ and $\mathbb{V}$ is the maximum of the directed distance between $\mathbb{U}$ and $\mathbb{V}$ and the directed distance between $\mathbb{V}$ and $\mathbb{U}$.

#### Keywords

metric, doubling measure, quasisymmetric map

#### 2010 Mathematics Subject Classification

Primary 54E35. Secondary 28A12, 51F99.

V.S. has been partly supported by the Academy of Finland via the Centre of Excellence in Analysis and Dynamics Research.

Received 3 September 2019

Received revised 13 April 2020

Accepted 28 April 2020

Published 3 November 2020