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MARTIN-LÖF RANDOMNESS IN SPACES OF CLOSED SETS

  • LOGAN M. AXON (a1)
Abstract

Algorithmic randomness was originally defined for Cantor space with the fair-coin measure. Recent work has examined algorithmic randomness in new contexts, in particular closed subsets of 2ɷ ([2] and [8]). In this paper we use the probability theory of closed set-valued random variables (RACS) to extend the definition of Martin-Löf randomness to spaces of closed subsets of locally compact, Hausdorff, second countable topological spaces. This allows for the study of Martin-Löf randomness in many new spaces, but also gives a new perspective on Martin-Löf randomness for 2ɷ and on the algorithmically random closed sets of [2] and [8]. The first half of this paper is devoted to developing the machinery of Martin-Löf randomness for general spaces of closed sets. We then prove some general results and move on to show how the algorithmically random closed sets of [2] and [8] fit into this new framework.

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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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