Real Analysis Exchange

On a Zero-Infinity Law of Olsen

Enrico Zoli

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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.

Article information

Real Anal. Exchange Volume 34, Number 1 (2008), 215-218.

First available in Project Euclid: 19 May 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 28A12: Contents, measures, outer measures, capacities 28A78: Hausdorff and packing measures
Secondary: 11J83: Metric theory

zero--infinity laws Hausdorff and packing measures scaling property Z-invariant sets translation-invariant metric measures


Zoli, Enrico. On a Zero-Infinity Law of Olsen. Real Anal. Exchange 34 (2008), no. 1, 215--218.

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