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.
"Uniform almost everywhere domination." J. Symbolic Logic 71 (3) 1057 - 1072, September 2006. https://doi.org/10.2178/jsl/1154698592