Schnorr randomness for noncomputable measures

TL;DR

This paper introduces a new definition of Schnorr randomness for noncomputable measures, establishing its properties and differences from Martin-Löf randomness through novel computable analysis techniques.

Contribution

It defines and validates a consistent notion of Schnorr randomness for noncomputable measures, expanding the theoretical framework of algorithmic randomness.

Findings

The new definition aligns with many properties of Martin-Löf randomness.

Proof techniques involve novel ideas from computable analysis.

The definition is shown to be correct and robust.

Abstract

This paper explores a novel definition of Schnorr randomness for noncomputable measures. We say $x$ is uniformly Schnorr $\mu$-random if $t(\mu,x)<\infty$ for all lower semicomputable functions $t(\mu,x)$ such that $\mu\mapsto\int t(\mu,x)\,d\mu(x)$ is computable. We prove a number of theorems demonstrating that this is the correct definition which enjoys many of the same properties as Martin-L\"of randomness for noncomputable measures. Nonetheless, a number of our proofs significantly differ from the Martin-L\"of case, requiring new ideas from computable analysis.