Measures and their random reals

Details Measures and their random reals Measures and their random reals

2008-02-19

arxiv.org/abs/0802.2705v1

Mathematics

Logic

Scientific paper

We study the randomness properties of reals with respect to arbitrary probability measures on Cantor space. We show that every non-recursive 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 a continuous measure can be found throughout the hyperarithmetical Turing degrees.

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