### Abstract

We study the randomness properties of reals with respect to arbitrary probability measures on Cantor space. We show that every noncomputable 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.

Original language | English (US) |
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Pages (from-to) | 5081-5097 |

Number of pages | 17 |

Journal | Transactions of the American Mathematical Society |

Volume | 367 |

Issue number | 7 |

DOIs | |

State | Published - Jul 1 2015 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

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## Cite this

Reimann, J. S., & Slaman, T. A. (2015). Measures and their random reals.

*Transactions of the American Mathematical Society*,*367*(7), 5081-5097. https://doi.org/10.1090/S0002-9947-2015-06184-4