Real Analysis Exchange

On the Steinhaus Property and Ergodicity via the Measure-Theoretic Density of Sets

Alexander Kharazishvili

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Abstract

It is shown how the Steinhaus property and ergodicity of a translation invariant extension \(\mu\) of the Lebesgue measure depend on the measure-theoretic density of \(\mu\)-measurable sets. Some connection of the Steinhaus property with almost convex sets is considered and a translation invariant extension of the Lebesgue measure is presented, for which the generalized Steinhaus property together with the mid-point convexity do not imply the almost convexity.

Article information

Source
Real Anal. Exchange, Volume 44, Number 1 (2019), 217-228.

Dates
First available in Project Euclid: 27 June 2019

Permanent link to this document
https://projecteuclid.org/euclid.rae/1561622441

Digital Object Identifier
doi:10.14321/realanalexch.44.1.0217

Mathematical Reviews number (MathSciNet)
MR3951343

Zentralblatt MATH identifier
07088972

Subjects
Primary: 28A05: Classes of sets (Borel fields, $\sigma$-rings, etc.), measurable sets, Suslin sets, analytic sets [See also 03E15, 26A21, 54H05] 28D05: Measure-preserving transformations
Secondary: 26A05

Keywords
Lebesgue measure Steinhaus property extension of measure, ergodicity, density point, mid-point convexity

Citation

Kharazishvili, Alexander. On the Steinhaus Property and Ergodicity via the Measure-Theoretic Density of Sets. Real Anal. Exchange 44 (2019), no. 1, 217--228. doi:10.14321/realanalexch.44.1.0217. https://projecteuclid.org/euclid.rae/1561622441


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