Archive for Mathematical Logic

, Volume 46, Issue 7–8, pp 533–546 | Cite as

Algorithmic randomness of continuous functions

  • George Barmpalias
  • Paul Brodhead
  • Douglas CenzerEmail author
  • Jeffrey B. Remmel
  • Rebecca Weber


We investigate notions of randomness in the space \({{\mathcal C}(2^{\mathbb N})}\) of continuous functions on \({2^{\mathbb N}}\). A probability measure is given and a version of the Martin-Löf test for randomness is defined. Random \({\Delta^0_2}\) continuous functions exist, but no computable function can be random and no random function can map a computable real to a computable real. The image of a random continuous function is always a perfect set and hence uncountable. For any \({y \in 2^{\mathbb N}}\), there exists a random continuous function F with y in the image of F. Thus the image of a random continuous function need not be a random closed set. The set of zeroes of a random continuous function is always a random closed set.


Computable analysis Computability Randomness 

Mathematics Subject Classification (2000)

03D28 68Q30 60D05 


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Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  • George Barmpalias
    • 1
  • Paul Brodhead
    • 2
  • Douglas Cenzer
    • 2
    Email author
  • Jeffrey B. Remmel
    • 3
  • Rebecca Weber
    • 4
  1. 1.School of MathematicsUniversity of LeedsLeedsUK
  2. 2.Department of MathematicsUniversity of FloridaGainesvilleUSA
  3. 3.Department of MathematicsUniversity of California at San DiegoLa JollaUSA
  4. 4.Department of MathematicsDartmouth CollegeHanoverUSA

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