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2008/2009 On a Zero-Infinity Law of Olsen
Enrico Zoli
Real Anal. Exchange 34(1): 215-218 (2008/2009).


Let $\mu$ be a translation-invariant metric measure on $\mathbb{R}$ with the following scaling property: for every $\lambda \in (0,1)$ there exists $b(\lambda)>\lambda$ with $\mu(\lambda X)\geq b(\lambda) \mu(X)$ for all $X \subseteq \mathbb{R}$. If $X$ is a $\mathbb{Z}$-invariant subset of $\mathbb{R}$ with $X/q \subseteq X$ for some $q\in \mathbb{N}\setminus \{1\}$, then $\mu(X)=0$ or $\mu(X\cap O)=\infty$ for every non-empty open set $O$. This refines an earlier result by Olsen.


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Enrico Zoli. "On a Zero-Infinity Law of Olsen." Real Anal. Exchange 34 (1) 215 - 218, 2008/2009.


Published: 2008/2009
First available in Project Euclid: 19 May 2009

zbMATH: 1184.28004
MathSciNet: MR2527134

Primary: 28A12 , 28A78
Secondary: 11J83

Keywords: Hausdorff and packing measures , Scaling property , translation-invariant metric measures , zero--infinity laws , Z-invariant sets

Rights: Copyright © 2008 Michigan State University Press

Vol.34 • No. 1 • 2008/2009
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