## Measure theory and higher order arithmetic

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## Abstract:

We investigate the statement that the Lebesgue measure defined on all subsets of the Cantor-space exists. As a base system we take $\textsf {ACA}_0^\omega +(\mu )$. The system $\textsf {ACA}_0^\omega$ is the higher order extension of Friedman’s system $\mathsf {ACA_0}$, and $(\mu )$ denotes Feferman’s $\mu$, that is, a uniform functional for arithmetical comprehension defined by $f(\mu (f))=0$ if $\exists {n} f(n)=0$ for $f\in \mathbb {N}^{\mathbb {N}}$. Feferman’s $\mu$ will provide countable unions and intersections of sets of reals and is, in fact, equivalent to this. For this reason $\mathsf {ACA_0}^\omega + (\mu )$ is the weakest fragment of higher order arithmetic where $\sigma$-additive measures are directly definable.

We obtain that over $\mathsf {ACA_0}^\omega + (\mu )$ the existence of the Lebesgue measure is $\Pi ^1_2$-conservative over $\mathsf {ACA_0}^\omega$ and with this conservative over $\mathsf {PA}$. Moreover, we establish a corresponding program extraction result.

## References

- Jeremy Avigad, Edward T. Dean, and Jason Rute,
*Algorithmic randomness, reverse mathematics, and the dominated convergence theorem*, Ann. Pure Appl. Logic**163**(2012), no. 12, 1854–1864. MR**2964874**, DOI 10.1016/j.apal.2012.05.010 - Jeremy Avigad and Solomon Feferman,
*Gödel’s functional (“Dialectica”) interpretation*, Handbook of proof theory, Stud. Logic Found. Math., vol. 137, North-Holland, Amsterdam, 1998, pp. 337–405. MR**1640329**, DOI 10.1016/S0049-237X(98)80020-7 - Jeremy Avigad, Philipp Gerhardy, and Henry Towsner,
*Local stability of ergodic averages*, Trans. Amer. Math. Soc.**362**(2010), no. 1, 261–288. MR**2550151**, DOI 10.1090/S0002-9947-09-04814-4 - Douglas K. Brown, Mariagnese Giusto, and Stephen G. Simpson,
*Vitali’s theorem and WWKL*, Arch. Math. Logic**41**(2002), no. 2, 191–206. MR**1890192**, DOI 10.1007/s001530100100 - Solomon Feferman,
*Theories of finite type related to mathematical practice*, Handbook of mathematical logic (Jon Barwise, ed.), North-Holland, Amsterdam, 1977, pp. 913–971. Studies in Logic and the Foundations of Math., Vol. 90. - Harvey Friedman,
*Systems of second order arithmetic with restricted induction, I, II*, J. Symbolic Logic**41**(1976), 557–559, abstract. - Harvey Friedman,
*A strong conservative extension of Peano arithmetic*, The Kleene Symposium (Proc. Sympos., Univ. Wisconsin, Madison, Wis., 1978), Studies in Logic and the Foundations of Mathematics, vol. 101, North-Holland, Amsterdam-New York, 1980, pp. 113–122. MR**591878** - Kurt Gödel,
*Über eine bisher noch nicht benützte Erweiterung des finiten Standpunktes*, Dialectica**12**(1958), 280–287 (German, with English summary). MR**102482**, DOI 10.1111/j.1746-8361.1958.tb01464.x - James Hunter,
*Higher-order reverse topology*, ProQuest LLC, Ann Arbor, MI, 2008. Thesis (Ph.D.)–The University of Wisconsin - Madison. MR**2711768** - Alexander P. Kreuzer and Ulrich Kohlenbach,
*Term extraction and Ramsey’s theorem for pairs*, J. Symbolic Logic**77**(2012), no. 3, 853–895. MR**2987141**, DOI 10.2178/jsl/1344862165 - U. Kohlenbach and L. Leuştean,
*A quantitative mean ergodic theorem for uniformly convex Banach spaces*, Ergodic Theory Dynam. Systems**29**(2009), no. 6, 1907–1915. MR**2563097**, DOI 10.1017/S0143385708001004 - S. C. Kleene,
*Recursive functionals and quantifiers of finite types. I*, Trans. Amer. Math. Soc.**91**(1959), 1–52. MR**102480**, DOI 10.1090/S0002-9947-1959-0102480-9 - Ulrich Kohlenbach,
*On the no-counterexample interpretation*, J. Symbolic Logic**64**(1999), no. 4, 1491–1511. MR**1780065**, DOI 10.2307/2586791 - Ulrich Kohlenbach,
*Higher order reverse mathematics*, Reverse mathematics 2001, Lect. Notes Log., vol. 21, Assoc. Symbol. Logic, La Jolla, CA, 2005, pp. 281–295. MR**2185441** - U. Kohlenbach,
*Applied proof theory: proof interpretations and their use in mathematics*, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2008. MR**2445721** - Alexander P. Kreuzer,
*On idempotent ultrafilters in higher-order reverse mathematics*, \ttfamily arXiv:1208.1424, accepted for publication in the Journal of Symbolic Logic. - Alexander P. Kreuzer,
*Non-principal ultrafilters, program extraction and higher-order reverse mathematics*, J. Math. Log.**12**(2012), no. 1, 1250002, 16. MR**2950192**, DOI 10.1142/S021906131250002X - Horst Luckhardt,
*Extensional Gödel functional interpretation. A consistency proof of classical analysis*, Lecture Notes in Mathematics, Vol. 306, Springer-Verlag, Berlin-New York, 1973. MR**0337512** - Jean Raisonnier,
*A mathematical proof of S. Shelah’s theorem on the measure problem and related results*, Israel J. Math.**48**(1984), no. 1, 48–56. MR**768265**, DOI 10.1007/BF02760523 - Gerald E. Sacks,
*Measure-theoretic uniformity in recursion theory and set theory*, Trans. Amer. Math. Soc.**142**(1969), 381–420. MR**253895**, DOI 10.1090/S0002-9947-1969-0253895-6 - B. Scarpellini,
*A model for barrecursion of higher types*, Compositio Math.**23**(1971), 123–153. MR**289257** - Noah Schweber,
*Transfinite Recursion in Higher Reverse Mathematics*, 2013, \ttfamily arXiv:1310.5792. - Saharon Shelah,
*Can you take Solovay’s inaccessible away?*, Israel J. Math.**48**(1984), no. 1, 1–47. MR**768264**, DOI 10.1007/BF02760522 - Stephen G. Simpson,
*Subsystems of second order arithmetic*, 2nd ed., Perspectives in Logic, Cambridge University Press, Cambridge; Association for Symbolic Logic, Poughkeepsie, NY, 2009. MR**2517689**, DOI 10.1017/CBO9780511581007 - Robert M. Solovay,
*A model of set-theory in which every set of reals is Lebesgue measurable*, Ann. of Math. (2)**92**(1970), 1–56. MR**265151**, DOI 10.2307/1970696 - Hisao Tanaka,
*A basis result for $\Pi _{1}{}^{1}$-sets of postive measure*, Comment. Math. Univ. St. Paul.**16**(1967/68), 115–127. MR**236017** - A. S. Troelstra (ed.),
*Metamathematical investigation of intuitionistic arithmetic and analysis*, Lecture Notes in Mathematics, Vol. 344, Springer-Verlag, Berlin-New York, 1973. MR**0325352** - Xiaokang Yu and Stephen G. Simpson,
*Measure theory and weak König’s lemma*, Arch. Math. Logic**30**(1990), no. 3, 171–180. MR**1080236**, DOI 10.1007/BF01621469

## Additional Information

**Alexander P. Kreuzer**- Affiliation: Department of Mathematics, Faculty of Science, National University of Singapore, Block S17, 10 Lower Kent Ridge Road, Singapore 119076
- Email: matkaps@nus.edu.sg
- Received by editor(s): June 7, 2014
- Received by editor(s) in revised form: October 27, 2014
- Published electronically: April 14, 2015
- Additional Notes: The author was partly supported by the RECRE project, and the Ministry of Education of Singapore through grant R146-000-184-112 (MOE2013-T2-1-062).
- Communicated by: Mirna Dz̆amonja
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**143**(2015), 5411-5425 - MSC (2010): Primary 03F35, 03B30; Secondary 03E35
- DOI: https://doi.org/10.1090/proc/12671
- MathSciNet review: 3411156