Computability of probability measures and Martin-Löf randomness over 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).

Schnorr randomness and the Lebesgue differentiation theorem

“…In Lemma 3.15 below we associate to each Methodologically, our proofs are perhaps somewhat novel. In verifying our Schnorr tests, we use Tarski's quantifier elimination theorem for the real number system (see Lemma 3.3 below) as well as some ideas from computable measure theory [12,15] (see Lemmas 2.12 and 3.5 below). So far as we know, this is the first time that quantifier elimination has been applied in randomness theory.…”

Applications of Effective Probability Theory to Martin-Löf Randomness

“…Definition 2.1.3 (from [HR09b]). A computable probability space is a pair (X, µ) where X is a computable metric space and µ is a computable Borel probability measure on X.…”

“…Its generalization to abstract spaces has been investigated in [ZL70,HW03,Gác05,HR09b]. We follow the approaches [Gác05,HR09b] developed on any computable probability space (X, µ).…”

“…This function is known to be the logarithm of a universal integrable µ-test, which means that for every integrable µ-test t there is a constant a such that log t ≤ a + d µ . On the other hand, every computable probability space admits a universal integrable test t µ (see [Gác05,HR09b]). Generalizing Davie, let us define K c := {x : t µ (x) ≤ 2 c }.…”

“…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.…”