Vitali Set Construction

Set Theory · Axiom Academy

EXAMPLE Constructing a Vitali Set Use the Axiom of Choice to construct a non-Lebesgue measurable set Excellent work! You've completed this example. Here's what we learned: Equivalence Relation: The relation x ~ y iff x - y ∈ ℚ partitions ℝ into uncountably many equivalence classes, each dense in ℝ. Axiom of Choice is Essential: We cannot explicitly describe which representatives to choose. We need AC to assert that such a choice function exists. Rational Translates: For each rational r, the set V + r = v + r : v ∈ V is a translate of V. These translates partition an interval around [0,1]. Contradiction from Measurability: If V were measurable, translation invariance would force its measure to be 0 (giving sum 0) or positive (giving sum ∞). Both contradict that the translates cover a bounded interval. Philosophical Impact: This shows that not every subset of ℝ can be assigned a reasonable "size". The Vitali set is pathological and requires AC to construct. The Vitali set is a cornerstone example showing the limits of Lebesgue measure and the surprising consequences of the Axiom of Choice!

This is the written version of the interactive lesson above. See the full Set Theory course.