Journal of Symbolic Logic

Uniform almost everywhere domination

Peter Cholak, Noam Greenberg, and Joseph S. Miller

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We explore the interaction between Lebesgue measure and dominating functions. We show, via both a priority construction and a forcing construction, that there is a function of incomplete degree that dominates almost all degrees. This answers a question of Dobrinen and Simpson, who showed that such functions are related to the proof-theoretic strength of the regularity of Lebesgue measure for Gδ sets. Our constructions essentially settle the reverse mathematical classification of this principle.

Article information

J. Symbolic Logic Volume 71, Issue 3 (2006), 1057-1072.

First available in Project Euclid: 4 August 2006

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Cholak, Peter; Greenberg, Noam; Miller, Joseph S. Uniform almost everywhere domination. J. Symbolic Logic 71 (2006), no. 3, 1057--1072. doi:10.2178/jsl/1154698592.

Export citation


  • W. Ackermann, Zum Hilbertschen Aufbau der reellen Zahlen., Mathematische Annalen, vol. 99 (1928), pp. 118--133.
  • Stephen Binns, Bjørn Kjos-Hanssen, Manuel Lerman, and Reed Solomon, On a conjecture of Dobrinen and Simpson concerning almost everywhere domination, 2005.
  • Natasha L. Dobrinen and Stephen G. Simpson, Almost everywhere domination, Journal of Symbolic Logic, vol. 69 (2004), no. 3, pp. 914--922.
  • Mariagnese Giusto and Stephen G. Simpson, Located sets and reverse mathematics, Journal of Symbolic Logic, vol. 65 (2000), no. 3, pp. 1451--1480.
  • Felix Hausdorff, Die Graduierung nach dem Endverlauf, Abhandlungen der Königlichen sächsischen Gesellschaft der Wissenschaften (Mathematisch-physische Klasse), vol. 31 (1909), pp. 296--334.
  • Carl G. Jockusch, Jr., Degrees of functions with no fixed points, Logic, methodology and philosophy of science, VIII (Moscow, 1987), Studies in Logic and the Foundations of Mathematics, vol. 126, North-Holland, Amsterdam, 1989, pp. 191--201.
  • Carl G. Jockusch, Jr. and Richard A. Shore, Pseudojump operators. I. The r.e. case, Transactions of the American Mathematical Society, vol. 275 (1983), no. 2, pp. 599--609.
  • Carl G. Jockusch, Jr. and Robert I. Soare, Degrees of members of $\Pi \sp0\sb1$ classes, Pacific Journal of Mathematics, vol. 40 (1972), pp. 605--616.
  • --------, $\Pi \sp0\sb1$ classes and degrees of theories, Transactions of the American Mathematical Society, vol. 173 (1972), pp. 33--56.
  • Antonín Kučera, Measure, $\Pi\sp 0\sb 1$-classes and complete extensions of $\rm PA$, Recursion theory week (Oberwolfach, 1984), Lecture Notes in Mathematics, vol. 1141, Springer, Berlin, 1985, pp. 245--259.
  • Stuart Kurtz Randomness and genericty in the degrees of unsolvability, Ph.D. thesis, University of Illinios at Urbana-Champaign, 1981.,
  • D. A. Martin, Classes of recursively enumerable sets and degrees of unsolvability, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 12 (1966), pp. 295--310.
  • André Nies, Lowness properties and randomness, Advances in Mathematics, vol. 197 (2005), pp. 274--305.
  • David B. Posner and Robert W. Robinson, Degrees joining to $\bf 0\sp\prime $, Journal of Symbolic Logic, vol. 46 (1981), no. 4, pp. 714--722.
  • Stephen G. Simpson Subsystems of second order arithmetic, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1999.,
  • S. Tennenbaum, Degree of unsolvability and the rate of growth of functions, Proceedings of the symposium on mathematical theory of automata (New York, 1962), Polytechnic Press of Polytechnic Institute of Brooklyn, Brooklyn, N.Y., 1963, pp. 71--73.
  • C. E. M. Yates, Three theorems on the degrees of recursively enumerable sets, Duke Mathematical Journal, vol. 32 (1965), pp. 461--468.
  • Xiaokang Yu and Stephen G. Simpson, Measure theory and weak König's lemma, Archive for Mathematical Logic, vol. 30 (1990), no. 3, pp. 171--180.