K-trivials Are Ncr

Definition 1.3. Suppose Z ∈ 2ω. A test relative to Z, or Z-test, is a set W ⊆ ω × 2<ω which is recursively enumerable in Z. For X ∈ 2ω, X passes a test W if and only if there is an n such that X 6∈ [Wn]. Definition 1.4. Suppose that m represents the measure μ on 2ω and that W is an m-test. • W is correct for μ if and only if for all n, ∑ σ∈Wn μ([σ]) ≤ 2 −n. • W is Solovay-correct for μ if and only if ∑ n∈ω μ([Wn]) <∞. Definition 1.5. X ∈ 2ω is 1-random relative to a representation m of μ if and only if X passes every m-test which is correct for μ. When m is understood, we say that X is 1-random relative to μ.