Randomness and halting probabilities

We consider the question of randomness of the probability ΩU [X] that an optimal Turing machine U halts and outputs a string in a fixed set X. The main results are as follows: • ΩU [X] is random whenever X is Σn-complete or Πn-complete for some n ≥ 2. • However, for n ≥ 2, ΩU [X] is not n-random when X is Σn or Πn. Nevertheless, there exists ∆n+1 sets such that ΩU [X] is n-random. • There are ∆2 sets X such that ΩU [X] is rational. Also, for every n ≥ 1, there exists a set X which is ∆n+1 and Σn-hard such that ΩU [X] is not random. We also look at the range of ΩU as an operator. We prove that the set {ΩU [X] : X ⊆ 2<ω} is a finite union of closed intervals. It follows that for any optimal machine U and any sufficiently small real r, there is a set X ⊆ 2 recursive in ∅′ ⊕ r, such that ΩU [X] = r. The same questions are also considered in the context of infinite computations, and lead to similar results.