The doubling metric and doubling measures
Flesch, Janos; Predtetchinski, Arkadi; Suomala, Ville (2020-11-03)
03.11.2020
Flesch, J., Predtetchinski, A., & Suomala, V. (2020). The doubling metric and doubling measures. Arkiv För Matematik, 58(2), 243–266. https://doi.org/10.4310/ARKIV.2020.v58.n2.a2
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Julkaisun pysyvä osoite on
https://urn.fi/URN:NBN:fi:oulu-202604282861
https://urn.fi/URN:NBN:fi:oulu-202604282861
Tiivistelmä
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 U of a metric space X, the predecessor U∗ of U is defined by doubling the radii of all open balls contained inside U, and taking their union. The predecessor of U is an open set containing U. The directed doubling distance between U and another subset V is the number of times that the predecessor operation needs to be applied to U to obtain a set that contains V . Finally, the doubling distance between open sets U and V is the maximum of the directed distance between U and V and the directed distance between V and U.
We introduce the so-called doubling metric on the collection of non-empty bounded open subsets of a metric space. Given an open subset U of a metric space X, the predecessor U∗ of U is defined by doubling the radii of all open balls contained inside U, and taking their union. The predecessor of U is an open set containing U. The directed doubling distance between U and another subset V is the number of times that the predecessor operation needs to be applied to U to obtain a set that contains V . Finally, the doubling distance between open sets U and V is the maximum of the directed distance between U and V and the directed distance between V and U.
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