On almost invariant subsets of the real line. (English) Zbl 1001.03047

Cichoń, Jacek; Szczepaniak, Przemyslaw

Acta Univ. Carol., Math. Phys. 42, No. 2, 17-21 (2001).

Summary: We show a new construction of almost invariant subsets of the real line. We first define almost invariant sets in some linear space and then we transport them by some special linear (over the field of rational numbers) isomorphism between this space and the real line treated as a linear space over the rationals. We show a construction of a non-measurable almost invariant subset of the real line and then we discuss the existence of Lebesgue measurable almost invariant sets.

MSC:

03E35	Consistency and independence results
28A05	Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets

Keywords:

almost invariant sets; Lebesgue measure; linear space; real line