Two problems on doubling measures
type: Item
instance of: scholarly article
title: Two problems on doubling measures
zbMATH Open document ID: 0862.28005
DOI: 10.4171/RMI/183
published in: Revista Matemática Iberoamericana
publication date: 1997-04-16T00:00:00Z
full work available at URL: https://eudml.org/doc/39490
review text: A measure \(\mu\) on a metric space \(X\) is doubling if \(\mu (B(x,2\rho)) \leq C \mu (B(x,\rho))\) for all open balls \(B(x,2\rho)\), \(B(x,\rho)\) and some \(C>0\). Doubling measures \(\neq 0\) exist on any compact metric space whose metric satisfies a condition of uniform branching, according to a theorem of Vol'berg and Konyagin. If a doubling measure on a compact metric space is continuous, there is a second one, singular with the first (a problem in the work cited). There are compact subsets of \(R^1\) carrying a purely atomic doubling measure and also one which is partly continuous. Besides doubling measures on \(R^1\) there are measures which are ``doubling'' for adjacent dyadic intervals whose union is also dyadic. There is a closed set \(S\) in \([0,1]\) which isn't a null set for dyadic doubling measures; but for almost all real \(t\), \(t+S\) is a null set for that class of measures.
Mathematics Subject Classification ID: 28A75
Mathematics Subject Classification ID: 54E45
Mathematics Subject Classification ID: 28C15
Mathematics Subject Classification ID: 30C62
Mathematics Subject Classification ID: 42B25
zbMATH DE Number: 838775
zbMATH Keywords: purely atomic doubling measures
zbMATH Keywords: dyadic doubling measures
author: Robert Kaufman
author: Jang-Mei Wu
MaRDI profile type: Publication
OpenAlex ID: W1992954021
description: scientific article; zbMATH DE number 838775