Schnorr randomness for noncomputable measures

Abstract This paper explores a novel definition of Schnorr randomness for noncomputable measures. We say x is uniformly Schnorr μ -random if t ( μ , x ) ∞ for all lower semicomputable functions t ( μ , x ) such that μ ↦ ∫ t ( μ , x ) d μ ( 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-Lof randomness for noncomputable measures. Nonetheless, a number of our proofs significantly differ from the Martin-Lof case, requiring new ideas from computable analysis.