Recursive events in random sequences

Abstract

Let ω be a Kolmogorov–Chaitin random sequence with ω1:n denoting the first n digits of ω. Let P be a recursive predicate defined on all finite binary strings such that the Lebesgue measure of the set {ω|∃nP(ω1:n)} is a computable real α. Roughly, P holds with computable probability for a random infinite sequence. Then there is an algorithm which on input indices for any such P and α finds an n such that P holds within the first n digits of ω or not in ω at all. We apply the result to the halting probability Ω and show that various generalizations of the result fail.