Randomness—beyond Lebesgue measure

Summary

Abstract. Much of the recent research on algorithmic randomness has focused on randomness for Lebesgue measure. While, from a computability theoretic point of view, the picture remains unchanged if one passes to arbitrary computable measures, interesting phenomena occur if one studies the the set of reals which are random for an arbitrary (continuous) probability measure or a generalized Hausdorff measure on Cantor space. This paper tries to give a survey of some of the research that has been done on randomness for non-Lebesgue measures.

Introduction. Most studies on algorithmic randomness focus on reals random with respect to the uniform distribution, i.e. the (1/2, 1/2)-Bernoulli measure, which is measure theoretically isomorphic to Lebesgue measure on the unit interval. The theory of uniform randomness, with all its ramifications (e.g. computable or Schnorr randomness) has been well studied over the past decades and has led to an impressive theory.

Recently, a lot of attention focused on the interaction of algorithmic randomness with recursion theory: What are the computational properties of random reals? In other words, which computational properties hold effectively for almost every real? This has led to a number of interesting results, many of which will be covered in a forthcoming book by Downey and Hirschfeldt [14].

While the understanding of “holds effectively” varied in these results (depending on the underlying notion of randomness, such as computable, Schnorr, or weak randomness, or various arithmetic levels of Martin-Löf randomness, to name only a few), the meaning of “for almost every” was usually understood with respect to Lebesgue measure.