On the Steinhaus property for invariant measures
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 \(\mathbb R\).
invariant extension of the Lebesgue measure
scientific article; zbMATH DE number 1095868