Randomness and Recursive Enumerability

One recursively enumerable real α dominates another one β if there are nondecreasing recursive sequences of rational numbers (a[n]:n∈ω) approximating α and (b[n]:n∈ω) approximating β and a positive constant C such that for all n, C(α-a[n])≥(β-b[n]). See [R. M. Solovay, Draft of a Paper (or Series of Papers) on Chaitin's Work, manuscript, IBM Thomas J. Watson Research Center, Yorktown Heights, NY, 1974, p. 215] and [G. J. Chaitin, IBM J. Res. Develop., 21 (1977), pp. 350--359]. We show that every recursively enumerable random real dominates all other recursively enumerable reals. We conclude that the recursively enumerable random reals are exactly the Ω-numbers [G. J. Chaitin, IBM J. Res. Develop., 21 (1977), pp. 350--359]. Second, we show that the sets in a universal Martin-Löf test for randomness have random measure, and every recursively enumerable random number is the sum of the measures represented in a universal Martin-Löf test.