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

Downey, Rod; Miller, Joseph

doi:10.1090/S0002-9939-05-07901-3

A basis theorem for $\Pi _1^0$ classes of positive measure and jump inversion for random reals
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by Rod Downey and Joseph S. Miller PDF

Proc. Amer. Math. Soc. 134 (2006), 283-288 Request permission

Abstract:

We extend the Shoenfield jump inversion theorem to the members of any $\Pi ^0_1$ class $\mathcal {P}\subseteq 2^\omega$ with nonzero measure; i.e., for every $\Sigma ^0_2$ set $S\geq _T\emptyset ’$, there is a $\Delta ^0_2$ real $A\in \mathcal {P}$ such that $A’\equiv _T S$. In particular, we get jump inversion for $\Delta ^0_2$ $1$-random reals.

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