Measures and their random reals

Jan Reimann, Theodore A. Slaman

Research output: Contribution to journalArticlepeer-review

15 Scopus citations


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 languageEnglish (US)
Pages (from-to)5081-5097
Number of pages17
JournalTransactions of the American Mathematical Society
Issue number7
StatePublished - Jul 1 2015

All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Applied Mathematics


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