Lowness notions, measure and domination

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 $K$

A Cappable Almost Everywhere Dominating Computably Enumerable Degree

AbstractWe show that there exists an almost everywhere (a.e.) dominating computably enumerable (c.e.) degree which is half of a minimal pair

Probabilistic vs Deterministic Gamblers

Can a probabilistic gambler get arbitrarily rich when all deterministic gamblers fail? We study this problem in the context of algorithmic randomness, introducing a new notion - almost everywhere computable randomness. A binary sequence X is a.e. computably random if there is no probabilistic computable strategy which is total and succeeds on X for positive measure of oracles. Using the fireworks technique we construct a sequence which is partial computably random but not a.e. computably random. We also prove the separation between a.e. computable randomness and partial computable randomness, which happens exactly in the uniformly almost everywhere dominating Turing degrees