arXiv:0802.2705 Abstract | arXiv Analytics

arXiv:0802.2705 [math.LO]Abstract References Reviews Resources

Measures and their random reals

Published 2008-02-19, updated 2013-05-14Version 2

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.

Categories: math.LO, math.PR

Subjects: 03D28, 68Q30

Keywords: random reals, continuous measure, arbitrary probability measures, randomness properties, cantor space

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