Algorithmically random closed sets and probability

by Logan M. Axon Algorithmic randomness in the Cantor space, 2 ω , has recently become the subject of intense study. Originally defined in terms of the fair coin measure, algorithmic randomness has since been extended, for example in Reimann and Slaman [22, 23], to more general measures. Others have meanwhile developed definitions of algorithmic randomness for different spaces, for example the space of continuous functions on the unit interval (Fouché [8, 9]), more general topological spaces (Hertling and Weihrauch [12]), and the closed subsets of 2 ω (Barmpalias et al. [1], Kjos-Hanssen and Diamondstone [14]). Our work has also been to develop a definition of algorithmically random closed subsets. We take a very different approach, however, from that taken by Barmpalias et al. [1] and Kjos-Hanssen and Diamondstone [14]. One of the central definitions of algorithmic randomness in Cantor space is Martin-Löf randomness. We use the probability theory of random closed sets (RACS) to prove that Martin-Löf randomness can be defined in the space of closed subsets of any locally compact, Hausdorff, second countable space. We then explore the Martin-Löf random closed subsets of the spaces N, 2 ω , and R under different measures. In the case of 2 ω we prove that the definitions of Barm-palias et al. [1] and Kjos-Hanssen and Diamondstone [14] are compatible with Logan M. Axon our approach. In the case of N we prove that the Martin-Löf random subsets are exactly those with Martin-Löf random characteristic functions. In the case of R we investigate the Martin-Löf random closed sets under generalized Poisson processes. This leads to a characterization of the Martin-Löf random elements of R as exactly the reals contained in some Martin-Löf random closed subset of R.