A stronger Kolmogorov zero-one law for resource-bounded measure

Abstract

Resource-bounded measure has been defined on the classes E, E/sub 2/, ESPACE, E/sub 2/SPACE, REC, and the class of all languages. It is shown here that if C is any of these classes and X is a set of languages that is closed under finite variations and has outer measure less than 1 in C, then X has measure 0 in C. This result strengthens Lutz's resource-bounded generalization of the classical Kolmogorov zero-one law. It also gives a useful sufficient condition for proving that a set has measure 0 in a complexity class.