Lowness notions, measure and domination (Q2890316)

In this much-anticipated paper, the authors give a complete answer to a problem posed by \textit{N. L. Dobrinen} and \textit{S. G. Simpson} [J. Symb. Log. 69, No. 3, 914--922 (2004; Zbl 1075.03021)] by characterising those Turing degrees which are \textit{almost everywhere dominating}, i.e., those degrees \(\mathbf{d}\) such that for almost all \(X\in 2^\omega\), all \(X\)-computable functions are dominated by some function in \(\mathbf{d}\). It is shown here that a degree \(\mathbf{d}\) is almost everywhere dominating if and only if every \(\mathbf{d}\)-random real is 2-random. This extends earlier results of the first author [Proc. Am. Math. Soc. 135, No. 11, 3703--3709 (2007; Zbl 1128.03031)] and of \textit{S. Binns} et al. [J. Symb. Log. 71, No. 1, 119--136 (2006; Zbl 1103.03014)]. The authors show that low-for-random reducibility \(\leqslant_{\mathrm{LR}}\) coincides with low-for-\(K\) reducibility \(\leqslant_{\mathrm{LK}}\), and that lowness for weak 2-randomness coincides with \(K\)-triviality. Proof-theoretic consequences regarding the strength of regularity of Lebesgue measure are also discussed.