Lowness notions, measure and domination

Abstract

We show that positive measure domination implies uniform almost everywhere domination and that this proof translates into a proof in the subsystem WWKL0 (but not in RCA0) of the equivalence of various Lebesgue measure regularity statements introduced by Dobrinen and Simpson. This work also allows us to prove that low for weak 2-randomness is the same as low for Martin-Löf randomness (a result independently obtained by Nies). Using the same technique, we show that⩽LR implies⩽LK, generalizing the fact that low for Martin-Löf randomness implies low for K.