An integral test for Schnorr randomness and its application

Some recent researches show that some classical theorems with \almost ev-erywhere " can be converted to eective version with \for each random point". One example is that a real is Martin-Lof random i each computable function of bounded variation is dierentiable at the real. The author [1] has given a Kurtz randomness version where we nd that an integral test is a useful tool to study the relation between algorithmic randomness and computable analysis. Here we give a version of Schnorr randomness. Consider a computable metric space (X; d; ) and a computable measure on it. De nition 1. An integral test for Schnorr randomness is a nonnegative lower semicomputable function t: X! R such that R td is a computable real. Theorem 1. A point z is Schnorr random i t(z) < 1 for each integral test t for Schnorr randomness