On the Steinhaus property for invariant measures. (English) Zbl 0879.28025

Kharazishvili, A. B.

Real Anal. Exch. 21(1995-96), No. 2, 743-749 (1996).

The well-known Steinhaus property of any set \(A\subset \mathbb R\) of strictly positive Lebesgue measure that the difference set \(A-A\) contains a neighborhood of \(0\) is studied. It is known that analogous property has any Haar measure on an arbitrary locally compact topological group. In the case of \(\mathbb R\) the property has an easy consequence: for every partition \(\{X,Y\}\) of \(\mathbb R\) consisting of Lebesgue measurable sets at least one of the difference sets \(X-X\), \(Y-Y\) contains the neighborhood of \(0\), but there are (non-measurable) partitions for which both of these difference sets have empty interiors. The author proved that still such a pathological partition can be constructed in a way that both sets \(X, Y\) are measurable with respect to an extension of the Lebesgue measure, which is invariant with respect to all isometric transformations of \(\mathbb R\).

Reviewer: J.Veselý (Praha)

MSC:

28C10	Set functions and measures on topological groups or semigroups, Haar measures, invariant measures
28A05	Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets

Keywords:

Steinhaus property; invariant measure; invariant extension of the Lebesgue measure; density point; Hamel basis