Lowness notions, measure and domination

TL;DR

This paper explores the relationships between measure domination, randomness notions, and logical subsystems, establishing equivalences and implications that deepen understanding of measure regularity and randomness in computability theory.

Contribution

It proves that positive measure domination implies uniform almost everywhere domination and translates this into a subsystem proof, also establishing equivalences between low randomness notions and implications between complexity measures.

Findings

Positive measure domination implies uniform almost everywhere domination.

Low for weak 2-randomness equals low for Martin-Löf randomness.

Low for Martin-Löf randomness implies low for K.

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-L\"of randomness (a result independently obtained by Nies). Using the same technique, we show that $\leq_{LR}$ implies $\leq_{LK}$, generalizing the fact that low for Martin-L\"of randomness implies low for $K$.