Cdmtcs Research Report Series Randomness Spaces Randomness Spaces

Martin-Lof de ned in nite random sequences over a nite 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, following a suggestion of Zvonkin and Levin. After studying basic results and constructions for such randomness spaces a general invariance result is proved which gives conditions under which a function between randomness spaces preserves randomness. This corrects and extends a result by Schnorr. Calude and J urgensen proved that the randomness notion for real numbers obtained by considering their b-ary representations is independent from the base b. We use our invariance result to show that this notion is identical with the notion which one obtains by viewing the real number space directly as a randomness space. Furthermore, arithmetic properties of random real numbers are derived, for example that every computable analytic function preserves randomness. Finally, 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. It is di erent from the usual one which is de ned via randomness of the characteristic function, but it can also be characterized in terms of random sequences. Surprisingly, it turns out that there are in nite co-r.e. random sets.