[21] Consider {0, 1} under the uniform measure. Let = 12 ... {0, 1}. (a) Show that if there is an infinite computable

[21] Consider {0, 1}∞ under the uniform measure. Let ω =

ω1ω2 ... ∈ {0, 1}∞.

(a) Show that if there is an infinite computable set I such that either for all i ∈ I we have ωi = 0 or for all i ∈ I we have ωi = 1, then ω is not random in the sense of Martin-L¨of.

(b) Show that if the set {i : ωi = 0} contains an infinite computably enumerable subset, then ω is not random in the sense of Martin-L¨of.

Comments. Source: [C. Calude and I. Chitescu, Ibid.].