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Effective Randomness of Unions and Intersections

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Abstract

We investigate the μ-randomness of unions and intersections of random sets under various notions of randomness corresponding to different probability measures. For example, the union of two relatively Martin-Löf random sets is not Martin-Löf random but is random with respect to the Bernoulli measure \(\lambda_{\frac{3}{4}}\) under which any number belongs to the set with probability \(\frac{3}{4}\). Conversely, any \(\lambda_{\frac{3}{4}}\) random set is the union of two Martin-Löf random sets. Unions and intersections of random closed sets are also studied.

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Authors and Affiliations

  1. Department of Mathematics, University of Florida, P.O. Box 118105, Gainesville, FL, 32611-8105, USA

    Douglas Cenzer

  2. Department of Mathematics, Dartmouth College, Hanover, NH, 03755, USA

    Rebecca Weber

Authors
  1. Douglas Cenzer
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  2. Rebecca Weber
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Correspondence to Douglas Cenzer.

Additional information

Research was partially supported by the National Science Foundation grants DMS 0532644 and 0554841 and 0062393. Thanks also to the American Institute of Mathematics for support during 2006 Effective Randomness Workshop.

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Cenzer, D., Weber, R. Effective Randomness of Unions and Intersections. Theory Comput Syst 52, 48–64 (2013). https://doi.org/10.1007/s00224-012-9416-1

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