a1 Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556-5683, USA. E-mail: Peter.Cholak.email@example.com
a2 Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556-5683, USA. E-mail: firstname.lastname@example.org
a3 Department of Mathematics, University of Connecticut, U-3009, 196 Auditorium Road, Storrs, CT 06269, USA. E-mail: email@example.com
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 G δ sets. Our constructions essentially settle the reverse mathematical classification of this principle.
(Received January 25 2006)