Real Analysis Exchange

Almost Isometry-Invariant Sets and Shadings

Keith Neu

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An almost isometry-invariant set $A\subset \mathbb{R}$ satisfies $|gA\ \Delta \ A| < \mathbf{c}$ for any isometry $g$ acting on $\mathbb{R}$, where ${\bf c}$ is the cardinality of the continuum. A shading is any set $S\subseteq \mathbb{R}$ in which $\frac{\mu(S\cap I)}{\mu(I)}$ has the same constant value for every finite interval $I$, for any Banach measure $\mu$. (A Banach measure is a finitely additive, isometry-invariant extension of the Lebesgue measure to $2^{\mathbb{R}}$.) In this paper we prove several theorems that show how these two types of sets are related. We also prove several sum and difference set results for almost isometry-invariant sets. Finally, we completely solve a problem involving subsets of Archimedean sets first posed by R. Mabry and partially solved by K. Neu.

Article information

Real Anal. Exchange, Volume 35, Number 2 (2009), 391-402.

First available in Project Euclid: 22 September 2010

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 28A12: Contents, measures, outer measures, capacities

almost invariance almost isometry-invariant almost translation-invariant Archimedean set shading Banach measure Hamel basis Lebesgue measure shade-almost invariance sum set difference set


Neu, Keith. Almost Isometry-Invariant Sets and Shadings. Real Anal. Exchange 35 (2009), no. 2, 391--402.

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