[38] Define 0

-lower semicomputable as lower semicomputable by a Turing machine equipped with an oracle for the halting problem, Section 1.7.2.

Here we use monotone Turing machines, since we want to deal with infinite sequences. Define 0 − μ-randomness of infinite binary sequences as previously μ-randomness, but this time using a 0

-lower semicomputable semimeasure M that is universal in the sense that it majorizes every 0

-lower semicomputable semimeasure. For every positive computable measure μ, every universal semimeasure M(x) (equivalently, universal reference monotonic Turing machine that generates it), and for every 0

−μ-random infinite sequence ω, there exists a finite limit of t(ω1:n), where t(x) = M(x)/μ(x).

Comments. This is a counterpart of Exercise 5.2.5; the counterpart of Exercise 5.2.6 is similar. Source: [An.A. Muchnik, Ibid.; M. Hutter and An.A. Muchnik, Ibid.].