Archive for Mathematical Logic 46 (7-8):533-546 (2008)

Abstract
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
Keywords Computable analysis  Computability  Randomness
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DOI 10.1007/s00153-007-0060-4
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References found in this work BETA

Arithmetical Representations of Brownian Motion I.Willem Fouché - 2000 - Journal of Symbolic Logic 65 (1):421-442.

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