Measures and their random reals

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...

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Bibliographic Details
Published in:Transactions of the American Mathematical Society 2015-07, Vol.367 (7), p.5081-5097
Main Authors: REIMANN, JAN, SLAMAN, THEODORE A.
Format: Article
Language:English
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Summary: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.
ISSN:0002-9947
1088-6850
DOI:10.1090/S0002-9947-2015-06184-4