Computability on the probability measures on the Borel sets of the unit interval

Abstract

While computability theory on many countable sets is well established and for computability on the real numbers several (mutually non-equivalent) definitions are applied, for most other uncountable sets, in particular for measures, no generally accepted computability concepts at all have been available until now. In this contribution we introduce computability on the set M of probability measures on the Borel subsets of the unit interval [0; 1]. Its main purpose is to demonstrate that this concept of computability is not merely an ad hoc definition but has very natural properties. Although the definitions and many results can of course be transferred to more general spaces of measures, we restrict our attention to M in order to keep the technical details simple and concentrate on the central ideas. In particular, we show that simple obvious reqirements exclude a number of similar definitions, that the definition leads to the expected computability results, that there are other natural definitions inducing the same computability theory and that the theory is embedded smoothly into classical measure theory. As background we consider TTE, Type 2 Theory of Effectivity [KW84, KW85], which provides a frame for very realistic computability definitions. In this approach, computability is defined on finite and infinite sequences of symbols explicitly by Turing machines and on other sets by means of notations and representations. Canonical representations are derived from information structures [Wei97]. We introduce a standard representation. δm: ⊆ ∑w→M via some natural information structure defined by a subbase σ (the atomic properties) of some topology τ on M and a standard notation of σ. While several modifications of δm suggesting themselves at first glance, violate simple and obvious requirements, δm 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 defined by a computable iterated function system with probabilities. Some other natural representations of M are introduced, among them a Cauchy representation associated with the Hutchinson metric, and proved to be equivalent to δm. As a corollary, the final topology τ of δm is the well known weak topology on M.