Computability of probability measures and Martin-Lof randomness over metric spaces

TL;DR

This paper extends the concept of algorithmic randomness to computable metric spaces, establishing a framework for computations with probability measures and demonstrating isomorphisms with the Cantor space.

Contribution

It develops a unified framework for computable probability measures on metric spaces and proves the existence of universal randomness tests in this setting.

Findings

Any computable metric space with a computable measure is isomorphic to the Cantor space.

Universal uniform randomness tests exist for all computable metric spaces.

Framework generalizes classical results from Cantor space to broader metric spaces.

Abstract

In this paper we investigate algorithmic randomness on more general spaces than the Cantor space, namely computable metric spaces. To do this, we first develop a unified framework allowing computations with probability measures. We show that any computable metric space with a computable probability measure is isomorphic to the Cantor space in a computable and measure-theoretic sense. We show that any computable metric space admits a universal uniform randomness test (without further assumption).