## Real Analysis Exchange

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

Alexander Kharazishvili

#### 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

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

#### 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