Schnorr randomness for noncomputable measures

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)dμ(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.