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The Reverse Mathematics of Analytic Measurability

Juan P. Aguilera, Thibaut Kouptchinsky, Keita Yokoyama

Abstract

A classical theorem of Lusin states that all analytic sets are Lebesgue-measurable. In this article we established the reverse mathematical strength of Lusin's theorem, which depends on how precisely it is formalized. By doing so, we answer to a question of Simpson. Our main proof is motivated towards proving a specific version of that result, namely that analytic sets are Lesbesgue-regular, which requires the equality of the outer and inner measures of the set in question. We prove this statement to be equivalent to $Σ^{1}_{1}$-$\mathrm{IND}$ over $\mathrm{ATR}_{0}$. The full statement of the theorem, that is the one implying the existence of the measure as a real number, is equivalent to $Π^{1}_{1}$-$\mathrm{CA}_{0}$, again provably over $\mathrm{ATR}_{0}$. In our main proof, we draw inspiration from Solovay's construction of a model of Zermelo-Fraenkel set theory where every set is Lebesgue measurable. In our case the argument requires the use of class forcing over a family of standard and non-standard models of a very weak set theory obtained through the method of pseudohierarchies.

The Reverse Mathematics of Analytic Measurability

Abstract

A classical theorem of Lusin states that all analytic sets are Lebesgue-measurable. In this article we established the reverse mathematical strength of Lusin's theorem, which depends on how precisely it is formalized. By doing so, we answer to a question of Simpson. Our main proof is motivated towards proving a specific version of that result, namely that analytic sets are Lesbesgue-regular, which requires the equality of the outer and inner measures of the set in question. We prove this statement to be equivalent to - over . The full statement of the theorem, that is the one implying the existence of the measure as a real number, is equivalent to -, again provably over . In our main proof, we draw inspiration from Solovay's construction of a model of Zermelo-Fraenkel set theory where every set is Lebesgue measurable. In our case the argument requires the use of class forcing over a family of standard and non-standard models of a very weak set theory obtained through the method of pseudohierarchies.
Paper Structure (13 sections, 21 theorems, 43 equations)

This paper contains 13 sections, 21 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and notation
  3. Models of set theory
  4. Wellorders
  5. Measure theory and Borel sets
  6. Reversals
  7. Proof of regularity
  8. The model $M$
  9. Class random forcing over non-standard models
  10. $M$-random reals
  11. Spector-Gandy representations
  12. Approximating sequences and nonstandard models
  13. Conclusions

Key Result

Theorem 1

The following are equivalent over ${\mathsf{ATR_0}}$.

Theorems & Definitions (37)

  • Theorem 1
  • Theorem 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • Lemma 8
  • Definition 9
  • Definition 10
  • ...and 27 more