Axiomatizing resource bounds for measure
CiE'11 Proceedings of the 7th conference on Models of computation in context: computability in Europe
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Lutz developed a general theory of resource-bounded measurability and measure on suitable complexity classes C⊆C (see Proceedings of the 13th IEEE Conference on Computational Complexity, pp. 236–248, 1998), where Cantor Space C is the class of all decision problems, and classes C include various exponential time and space complexity classes, the class of all decidable languages, and the Cantor space C itself. In this paper, a different general theory of resource-bounded measurability and measure on those complexity classes is developed. Our approach is parallel to the Carathéodory outer measure approach in classical Lebesgue measure theory. We shall show that many nice properties in the classical Lebesgue measure theory hold in the resource-bounded case also. The Carathéodory approach gives short and easy proofs of theorems in the resource-bounded case as well as in the classical case. The class of measurable sets in our paper is strictly larger than that of Lutz, and the two resource-bounded measures assign the same measure for a set if the set is measurable in the sense of Lutz.