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October 2020 The doubling metric and doubling measures
János Flesch, Arkadi Predtetchinski, Ville Suomala
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Ark. Mat. 58(2): 243-266 (October 2020). DOI: 10.4310/ARKIV.2020.v58.n2.a2

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 $\mathbb{U}$.

Funding Statement

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

Citation

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János Flesch. Arkadi Predtetchinski. Ville Suomala. "The doubling metric and doubling measures." Ark. Mat. 58 (2) 243 - 266, October 2020. https://doi.org/10.4310/ARKIV.2020.v58.n2.a2

Information

Received: 3 September 2019; Revised: 13 April 2020; Published: October 2020
First available in Project Euclid: 16 January 2021

Digital Object Identifier: 10.4310/ARKIV.2020.v58.n2.a2

Subjects:
Primary: 54E35
Secondary: 28A12, 51F99

Rights: Copyright © 2020 Institut Mittag-Leffler

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Vol.58 • No. 2 • October 2020
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