Intuitively, a random real should pass every feasible statistical test. This is the motivation for using measure theory to define randomness. In the language of measure theory, a test is associated with a null set. The following is a summary of the key notions of measure-theoretic randomness.
The notion of (now called) Kurtz randomness was introduced in Kurtz .
(i)A set A ⊆ 2ω is a Kurtz test if it is and μ(A) = 0;
(ii)a real x is Kurtz random if x ∉ ...