Computability on the Probability Measures on the Borel Sets of the Unit Interval

Scientists apply digital computers to perform computations on natural numbers, nite strings, real numbers and more general objects like sets, functions and measures. While computability theory on many countable sets is well established and for computability on the real numbers several (unfortunately mutually non-equivalent) deenitions are being studied, in particular for measures no computability concept at all has been available until now. In this contribution we introduce a natural computability concept on the set M of probability measures on the Borel subsets of the unit interval 0; 1]. As background we consider TTE, Type 2 Theory of EEectivity, KW84, KW85], where computability is deened on nite and innnite sequences of symbols explicitly by Turing machines and on other sets by means of notations and representations. A standard representation m : ! ?! M is introduced via some natural information structure Wei95a] (M; ;), where is a subbase of some T 0-topology m and : ?! is a standard notation of. While several modiications of m suggesting itself at rst glance have undesirable properties, m itself has several very natural properties and hence should induce an important computability theory. Many interesting functions on measures turn out to be computable, in particular linear combination , integration of continuous functions and any transformation deened by a computable iterated function system with probabilities. Some other representations of M are introduced, among them a Cauchy representation associated with the Hutchinson metric, and proved to be equivalent to m. In particular, the nal topology m of m is the well known weak topology on M.