The doubling metric and doubling measures

TL;DR

This paper introduces the doubling metric on open subsets of metric spaces, measuring how many doubling steps are needed to cover one set with another, providing a new way to analyze geometric properties of measures.

Contribution

The paper defines a novel doubling metric based on the predecessor operation, offering a new tool for studying the structure of doubling measures and metric space geometry.

Findings

Defines the doubling metric and predecessor operation.

Establishes properties of the doubling distance between sets.

Provides a framework for analyzing doubling measures geometrically.

Abstract

We introduce the so--called doubling metric on the collection of non--empty bounded open subsets of a metric space. Given a 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. If $U$ is open, 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 $U$ and $V$ is the maximum of the directed distance between $U$ and $V$ and the directed distance between $V$ and $U$.