Randomness spaces

Abstract

Martin-Löf defined infinite random sequences over a finite alphabet via randomness tests which describe sets having measure zero in a constructive sense. In this paper this concept is generalized to separable topological spaces with a measure. We show several general results, like the existence of a universal randomness test under weak conditions, and a randomness preservation result for functions between randomness spaces. Applying these ideas to the real numbers yields a direct definition of random real numbers which is shown to be equivalent to the usual one via the representation of real numbers to some base. Furthermore, we show that every nonconstant computable analytic function preserves randomness. As a second example, by considering the power set of the natural numbers with its natural topology as a randomness space, we introduce a new notion of a random set of numbers. We characterize it in terms of random sequences. Surprisingly, it turns out that there are infinite co-r.e. random sets.