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-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.