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Measures and their random reals


Authors: Jan Reimann and Theodore A. Slaman
Journal: Trans. Amer. Math. Soc. 367 (2015), 5081-5097
MSC (2010): Primary 03D32, 68Q30
DOI: https://doi.org/10.1090/S0002-9947-2015-06184-4
Published electronically: January 30, 2015
MathSciNet review: 3335411
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Abstract: We study the randomness properties of reals with respect to arbitrary probability measures on Cantor space. We show that every non-computable real is non-trivially random with respect to some measure. The probability measures constructed in the proof may have atoms. If one rules out the existence of atoms, i.e. considers only continuous measures, it turns out that every non-hyperarithmetical real is random for a continuous measure. On the other hand, examples of reals not random for any continuous measure can be found throughout the hyperarithmetical Turing degrees.


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Additional Information

Jan Reimann
Affiliation: Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania 16802
Email: reimann@math.psu.edu

Theodore A. Slaman
Affiliation: Department of Mathematics, University of California at Berkeley, Berkeley, California 94720
Email: slaman@math.berkeley.edu

DOI: https://doi.org/10.1090/S0002-9947-2015-06184-4
Received by editor(s): February 19, 2008
Received by editor(s) in revised form: May 14, 2013
Published electronically: January 30, 2015
Additional Notes: The first author was partially supported by NSF grants DMS-0801270 and DMS-1201263.
The second author was partially supported by NSF grants DMS-0501167 and DMS-1001551.
Article copyright: © Copyright 2015 American Mathematical Society