Measure-theoretic uniformity and the Suslin functional

Abstract

Given a set A in the unit interval and the associated Lebesgue measure λ, it is a natural question whether we may (in some sense) compute the measure λ ( A ) in terms of the set A. Under the moniker measure theoretic uniformity, Tanaka and Sacks have (independently) provided a positive answer for the well-known class of hyperarithmetical sets of reals, and provided a basis theorem for such sets of positive measure. The hyperarithmetical sets are exactly the sets computable in terms of the functional 2 E, in the sense of Kleene’s S1–S9. In turn, Kleene’s 2 E essentially corresponds to arithmetical comprehension as in ACA 0 . In this paper, we generalise the aforementioned results to the ‘next level’, namely Π 1 1 -CA 0 , in the form of the Suslin functional, or the equivalent hyperjump. We also generalise the Tanaka-Sacks basis theorem to sets of positive measure that are semi-computability relative to the Suslin functional. Finally, we discuss similar generalisations for infinite time Turing machines.