Separations by Random Oracles and "Almost" Classes for Generalized Reducibilities

Given two binary relations ⩽r and ⩽s on 2ω which are closed under finite variation (of their set arguments) and a set X chosen randomly by independent tosses of a fair coin, one might ask for the probability that the lower cones \(\left\{ {A \subseteq \omega :A \leqslant _r X} \right\} and \left\{ {A \subseteq \omega :A \leqslant _3 X} \right\}\)w.r.t. ⩽r and ⩽s are different. By closure under finite variation, the Kolmogorov 0–1 Law yields immediately that this probability is either 0 or 1; in the case it is 1, the relations are said to be separable by random oracles. Again by closure under finite variation, the probability that a randomly chosen set X is in the upper cone of a fixed set A w.r.t. ⩽r is either 0 or 1. Almost r is the class of sets for which the upper cone w.r.t. ⩽r has measure 1.