Measures and measurable sets

Construction and uniqueness of measures, atomless and vector measures, measures on Boolean algebras, and the limits of measurability shown by non-measurable sets and paradoxical decompositions.

• Atomless measures
• Banach–Tarski paradox
• Carathéodory's extension theorem
• Congruence-invariant extensions of Lebesgue measure
• Independent bases of atomless measures
• Lyapunov convexity theorem
• Measures on complete Boolean algebras
• Uniqueness of measures on Euclidean space

TheoremSzpilrajn's Problem 170: plane sets with measurable homeomorphic copies#●ready to prove

Let \(S\) be a subset of the plane \(\mathbb{R}^2\) such that every subset \(T\) of \(\mathbb{R}^2\) homeomorphic to \(S\) (as a subspace) is Lebesgue measurable, that is, Carathéodory measurable with respect to Lebesgue outer measure on \(\mathbb{R}^2\). Then \(S\) is absolutely measurable: for every measure \(\mu\) on the Borel \(\sigma\)-algebra of \(\mathbb{R}^2\), the set \(S\) is Carathéodory measurable with respect to the outer measure induced by \(\mu\).

The hypothesis concerns every subset of the plane homeomorphic to \(S\), not only the images of \(S\) under homeomorphisms of the whole plane.

theorem_wanted szpilrajn170_homeomorphic_images_measurable_imp_absolute :
    ∀ S : Set (ℝ × ℝ),
      (∀ T : Set (ℝ × ℝ), Nonempty (↥S ≃ₜ ↥T) →
        @MeasurableSet (ℝ × ℝ) (MeasureTheory.volume.toOuterMeasure.caratheodory) T) →
      ∀ μ : MeasureTheory.Measure (ℝ × ℝ),
        @MeasurableSet (ℝ × ℝ) (μ.toOuterMeasure.caratheodory) S

Sources

• E. Szpilrajn, Problem 170 of the Scottish Book; the affirmative answer follows from an unpublished result of J. von Neumann, as recorded in Ad 170 of the typescript. See the commentary in R. D. Mauldin (ed.), The Scottish Book, 2nd ed. (Birkhäuser, 2015). Lebesgue measurability is stated as Carathéodory measurability with respect to volume; the case of linear sets, and the analogue for generalized homeomorphisms in the sense of Kuratowski, were already known.

TheoremVitali's theorem#✓fully proved

There exists a subset of \(\mathbb{R}\) that is not Lebesgue measurable.

theorem vitali_nonmeasurable :
    ∃ s : Set ℝ, ¬ NullMeasurableSet s (volume : Measure ℝ)

Sources

• G. Vitali, Sul problema della misura dei gruppi di punti di una retta, Tipografia Gamberini e Parmeggiani, Bologna (1905). Here NullMeasurableSet w.r.t. volume is the Lebesgue-measurability notion, i.e. measurability for the completion of Lebesgue measure.

1 fully proved1 ready to prove

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