Exercise · ZW-VITALI-SET-007 · ★★☆ Intermediate
If \(V\) is a Vitali set, what does the union \(\bigcup_n (V+q_n)\) contain for an enumeration \(q_n\) of \(\mathbb{Q}\cap[-1,1]\)?
Choose one option.
Show answer
- A. The interval \([0,1]\)
Explanation
Every real in \([0,1]\) is congruent modulo \(\mathbb{Q}\) to an element of \(V\), and that element appears in some translate. The union is not just \(V\) nor an empty set.
Review the concept: Vitali set
The Vitali set is a subset of \([0,1]\) containing exactly one representative of each class of \(\mathbb{R}\) modulo \(\mathbb{Q}\), and it is not Lebesgue measurable.
\[x \sim y \iff x - y \in \mathbb{Q}\]
More practice problems: vitali set
- ZW-VITALI-SET-001 Which axiom is used to choose a representative from each equivalence class in Vitali's construction?
- ZW-VITALI-SET-002 What equivalence relation is used to construct the Vitali set?
- ZW-VITALI-SET-003 Which property of Lebesgue measure is used in Vitali's contradiction?
- ZW-VITALI-SET-004 What is the Lebesgue measure of the interval [0,1]?
- ZW-VITALI-SET-005 How many rationals in [-1,1] are used to translate the Vitali set?
- ZW-VITALI-SET-006 If V is a Vitali set and q∈ℚ, what is the Lebesgue measure of V+q when m(V)=0?
- ZW-VITALI-SET-008 If m(V)>0 and the translates V+q_n are disjoint, what follows about the measure of their union?
- ZW-VITALI-SET-009 What is the error in claiming the Cantor set is non-measurable because it is uncountable?