  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 $G_\delta$ sets. Our constructions essentially settle the reverse
mathematical classification of this principle.
  Revised Nov 13, 2005. Minor corrections made.
