Some properties of Vitali sets

Abstract

Abstract If V 1 , … , V n {V_{1},\dots,V_{n}} are translations of a Vitali set by rational numbers, then we prove that ⋃ i = 1 n V i {\bigcup_{i=1}^{n}V_{i}} contains no measurable subset of positive measure. This provides a decomposition of ℝ {{\mathbb{R}}} as a countable union of disjoint sets, any finite union of which has Lebesgue inner measure zero. As a consequence, we present a function δ : ℝ → ( 0 , + ∞ ) {\delta:{\mathbb{R}}\rightarrow(0,+\infty)} for which there is no measurable function f : ℝ → ℝ {f:{\mathbb{R}}\rightarrow{\mathbb{R}}} satisfying 0 < f ≤ δ {0<f\leq\delta} , on any measurable set of positive Lebesgue measure.