[15] Show that an infinite sequence is random with respect to a computable measure iff the probability ratio (1:n)/M(1:n) is bounded below. Comments.

[15] Show that an infinite sequence ω is random with respect to a computable measure μ iff the probability ratio μ(ω1:n)/M(ω1:n) is bounded below.

Comments. Hint: see Theorem  4.5.7.

This ratio can be viewed as the likelihood ratio of hypothesis μ(x) and the fixed alternative hypothesis M(x). Source: [L.A. Levin, Soviet Math. Dokl., 14(1973), 1413–1416].