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 WWKL${}_0$ (but not in RCA${}_0$) 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 ${\le }_{LR}$ implies ${\le }_{LK}$, generalizing the fact that low for Martin-Löf randomness implies low for $K$.

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