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A basis theorem for $\Pi_1^0$ classes of positive measure and jump inversion for random reals

Authors: Rod Downey and Joseph S. Miller
Journal: Proc. Amer. Math. Soc. 134 (2006), 283-288
MSC (2000): Primary 03D28, 68Q30
Published electronically: August 11, 2005
MathSciNet review: 2170569
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Abstract: We extend the Shoenfield jump inversion theorem to the members of any $\Pi^0_1$class $\+ P\subseteq 2^\omega$ with nonzero measure; i.e., for every $\Sigma^0_2$ set $S\geq_T\emptyset'$, there is a $\Delta^0_2$ real $A\in\+ P$such that $A'\equiv_T S$. In particular, we get jump inversion for $\Delta^0_2$$1$-random reals.

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Additional Information

Rod Downey
Affiliation: School of Mathematical and Computing Sciences, Victoria University, P.O. Box 600, Wellington, New Zealand

Joseph S. Miller
Affiliation: Department of Mathematics, University of Connecticut, U-3009, 196 Auditorium Road, Storrs, Connecticut 06269

Received by editor(s): April 13, 2004
Received by editor(s) in revised form: June 30, 2004
Published electronically: August 11, 2005
Additional Notes: Both authors were supported by the Marsden Fund of New Zealand. The first author was also partially supported by NSFC Grand International Joint Project Grant No. 60310213 “New Directions in Theory and Applications of Models of Computation” (China).
Communicated by: Carl G. Jockusch, Jr.
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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