A basis theorem for Π₁⁰ classes of positive measure and jump inversion for random reals

Abstract

We extend the Shoenfield jump inversion theorem to the members of any Π 1 0 \Pi ^0_1 class P ⊆ 2 ω \mathcal {P}\subseteq 2^\omega with nonzero measure; i.e., for every Σ 2 0 \Sigma ^0_2 set S ≥ T ∅ ′ S\geq _T\emptyset ’ , there is a Δ 2 0 \Delta ^0_2 real A ∈ P A\in \mathcal {P} such that A ′ ≡ T S A’\equiv _T S . In particular, we get jump inversion for Δ 2 0 \Delta ^0_2 1 1 -random reals.