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

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.