An Application of Martin-Löf Randomness to Effective Probability Theory

Abstract

Abstract. In this paper we provide a framework for computable analysis of measure, probability and integration theories. We work on computable metric spaces with computable Borel probability measures. We introduce and study the framework of layerwise computability which lies on Martin-Löf randomness and the existence of a universal randomness test. We then prove characterizations of effective notions of measurability and integrability in terms of layerwise computability. On the one hand it gives a simple way of handling effective measure theory, on the other hand it provides powerful tools to study Martin-Löf randomness, as illustrated in a sequel paper.

Applications of Effective Probability Theory to Martin-Löf Randomness

“…In this new framework, which we call layerwise computability, the layerwise versions of virtually all computability notions can be naturally defined. The contributions of this setting can be summarized in the following principle, supported by the main results in [HR09a]:…”

“…The following is an adaptation of [HR09a] to complete spaces. Let [S] µ be the set of Borel subsets of X quotiented by the equivalence relation…”

“…2.1.5). It was demonstrated in [HR09a] that algorithmic randomness and the universal test offer an alternative elegant way of handling effective measurability notions. Let (X, µ) be a complete computable probability space.…”

“…In [HR09a], working in the context of computable probability spaces (to which Martin-Löf randomness has been recently extended, see [Gác05,HR09b]), effective versions of measure-theoretic notions were examined and another contribution of algorithmic randomness to probability theory was developed: the setting of a new framework for computability adapted to the probabilistic context. This was achieved by making a fundamental use of the existence of a universal Martin-Löf test to endow the space with what we call the Martin-Löf layering.…”

“…In this paper, elaborating on [HR09a] and developing layerwise computability further, we give solutions to Problems 1 and 2. The CP is at the core of these solutions, that we briefly present now.…”