2008/02/19 by Jan Reimann, Reimann, Jan, Theodore A. Slaman +1 · 1 citation
Computer Science · Mathematics · #03D28 #68Q30 #Advanced Topology and Set Theory #Cellular Automata and Applications #Computability, Logic, AI Algorithms #FOS: Mathematics #Logic (math.LO) #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.0802.2705
openalex publication_date 2008/02/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.