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:
[start-list]
* ΩU[X] is random whenever X is Σ⁰n-complete
or Π⁰n-complete for some n≥2.
* However, for n≥2, ΩU[X] is not n-random
when X is Σ⁰n or Π⁰n.
Nevertheless, there exists Δ⁰n+1 sets such that
ΩU[X] is n-random.
* There are Δ⁰₂ sets X such that ΩU[X] is rational.
Also, for every n≥1, there exists a set X which is
Δ⁰n+1 and Σ⁰n-hard
such that ΩU[X] is not random.
[end-list]
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< ω recursive in ∅’⊕ r, such that ΩU