Schnorr randomness for noncomputable measures

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-Löf randomness for noncomputable measures. Nonetheless, a number of our proofs significantly differ from the Martin-Löf case, requiring new ideas from computable analysis.